中英文文献翻译静电悬浮磁盘的水平系统仿真.docx
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中英文文献翻译静电悬浮磁盘的水平系统仿真.docx
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中英文文献翻译静电悬浮磁盘的水平系统仿真
SystemLevelSimulationofanElectrostaticallyLevitatedDisk
MichaelKraftandAlanEvans
UniversityofSouthampton,Highfield,Southampton,SO171BJ
ABSTRACT
Thispaperdescribesthederivationofasystem-levelmodelforamicromachineddiskwhichislevitatedbyelectrostaticforces.Suchasystemhasmanypotentialapplicationsforinertialsensors,micro-motors,microfluidicandmicro-opticaldevices.AsasimulationtoolMatlab/Simulinkwasusedsinceitcaneasilyhandledifferentdomainssuchaselectricalandmechanicalwithoutspecialarrangements.Multi-axiselectro-mechanicalsigmadeltamodulatorswerechosentocontrolthevariousdegreesoffreedomofthelevitateddisk.Ofspecialinterestisthepower-upphaseattheendofwhichthecontrolsystemhastoensurethatthediskiscentredbetweenthetopandbottomelectrodes.
Keywords:
Electrostaticlevitation,inertialsensors,systemlevelsimulation.
1INTRODUCTION
Electrostaticforcesarecommonlyusedforactuationofmicromachineddevicese.g.force-balancedinertialsensors,microfluidicvalvesandmicro-opticalmirrors.However,inallthesedevicesamechanicalconnectionexistsbetweentheactuatedpartandthesubstrate,thepropertiesofwhicharesubjecttoconsiderableprocesstolerancesandcannotbechangedeasilyoncethedevicehasbeenmanufactured.Inthispaperamicromachineddiskissuggestedwhichislevitatedbyelectrostaticforces.Thishasmanypotentialapplicationssuchasaccelerometerswithonlinedynamiccharacteristicstuning,gyroscopesifthediskisspunandtheprecessionofthediskinducedbytheCoriolisforceismeasured,microfluidicmixersandpumps,frictionlessbearingsformicro-motorsandmicro-opticallightchoppers.Surprisinglylittleworkhasbeenundertakeninthisdirection.
Thesystem,asshowninfig.1,consistsofanickeldiskmanufacturedbyelectroplatingwhichisencagedbysetsofelectrodesontopandbottomandelectroplatedpillarsatthesides.Themanufacturingprocesswillbedescribedelsewhereindetail.Eachsetofelectrodesformsfivecapacitorswhichareusedforbothsensingthepositionofthediskaswellasactuatingitinsuchawaythatitismaintainedatorclosetothecentreposition.
Inthisworkthesystemlevelsimulationofsuchalevitateddiskispresented.Simulinkwaschosenasasimulationtoolsinceitisidealforthesimulationofamicromachinedsystembecauseiteasilycanhandledifferentdomainssuchaselectricalandmechanicalwithoutspecialarrangements.Thesystemisunstableintheopenloopconfiguration,consequently,thecontrolstrategyhastobeconsideredwithgreatcare.Here,multi-axiselectromechanicalsigma-deltamodulatorsareused[3]tocontrolthevariousdegreesoffreedomofthedisk.Thispreventsthepossibilityofelectrostaticlatch-up[4]andalsoresultsinaninherentlydigitalsystem.
2DERIVATIONOFTHEMODEL
Inordertoderiveamodelcapturingthegoverningfeaturesofthesystemthefollowingbuildingblockshavetobeconsidered:
-Thedynamicbehaviourofthedisk,
-conversionfromthemechanicaltotheelectricaldomain,i.e.thepositionmeasurementinterface,
-themulti-axiselectromechanicalsigma-deltamodulator,
-thecompensator,
-theresetmechanism,i.e.conversionfromthefeedbackvoltagetotheelectrostaticforcesandmomentsincludingtheeffectsofthecurrentpositionofthedisk.
2.1DiskDynamics
Thedynamicbehaviourofthelevitateddiskischaracterizedbythreesecondorderdifferentialequations:
wheremisthemassofthedisk,zthedisplacementfromthemid-positionbetweentheelectrodes,andtheangulardeflectionsaboutthexandyaxisofthedisk,bz,bandbthedampingcoefficientinz,anddirection
respectively,IxandIythemomentsofinertiaofthediskaboutthecorrespondingaxisandFext,z,Mext,xandMext,ytheexternalforceandmomentsinthecorrespondingdirections.
Themaindifferencetotheusualdynamicequationsdescribingamicromachinedproofmassisthatamechanicalspringforcetermismissing.Theeffectivespringconstantofthediskisentirelycontrolledbytheexternalelectrostaticforcesandmoments.Asadrawbackitobviousthatthesystemcannotoperateinanopenloopmode,asusuallypossibleforinertialsensors.Afurtherbuildingblockisrequiredtosimulatethephysicalconstraintofthedisk,thisisachievedbysaturationblocks.
2.2PositionMeasuringInterface
Thepositionofthedisk,characterizedbyitsz,andcoordinates,hastobemeasured.Thisisachievedbymeasuringthedifferentialcapacitanceofthefourouter,pieshapedelectrodes.Thesystemlevelmodeldoesnotsimulateanelectronicimplementation,however,itisenvisagedtoapplyanexcitationvoltagetotheround,middleelectrodesandsensethecurrentsfromtheouterelectrodesfromwhichitpossibletomeasurethefourdifferentialcapacitancesbetweenthecorrespondingtopandbottomelectrodes.
Theinformationaboutz,anddeflectionsisencodedinthedifferentialcapacitancevalues.Twopossiblecontrolapproachesarethereforepossible.Firstly,onecouldextractthepositioninformationatthispointofthecontrolloopwhichwouldresultinathree-axissigma-deltamodulatorcontrolscheme.Secondly,asigma-deltaloopwithfourpathsispossible,oneforeachdifferentialcapacitance.Theinformationaboutdiskpositionandexternalinertialforcesandmomentsisthenencodedinthedigitalbitstreamprovidingtheoutputsignalsofthesystem.Thisapproachwaschosenheresincethedecodingofthepositioninformationcanthenbeundertakeninthedigitaldomainwithobviousadvantages.Forthesystemlevelmodelananalyticalexpressionforthecapacitancebetweenthepie-shapedelectrodesandthediskhastobederived.Theexpressionforthisgeometryisverylengthyanditwasdecidedtouseanapproximationforasquareelectrode.Asanexampletheexpressionforthetop-rightsegmentisgiven:
whereε0isthedielectricconstantofvacuum(assumedtobethesameasforair),Risthediskradius,z0thenominaldistancebetweendiskandelectrodeswhenthediskisatmid-position.Furthermore,smallangulardisplacementsareassumedsothattheusualapproximationofparallel-platecapacitorsismade.Theexpressionsfortheotherelectrodescanbederivedbysymmetryconsiderations.Theuseoftheequationisillustratedinfig.2a.Thissquareplateapproximationisjustifiedbythefactthatintheforwardpathofthesigma-deltamodulatorcontrolsystemanidealcomparatorislocatedwhoseoutputisnotdeterminedbyanumericalexactsolutionbutrathergovernedbythesignofitsinputsignal.Itshouldbenotedthattheexpressionforthecapacitancehasasingularityat=0or=0,whichcausesasimulationerroratzerodeflectionandincorrectnumericalresultsforverysmallanglesduetonumericalnoise.Thisproblemcanbeovercomebyusingalinearizedversionofeq.
(2)around=0and=0,bothexpressionsarethenimplementedinthemodel,dependingonthemagnitudeoftheangulardisplacementseitherthelinerizedorfullexpressionisused.
2.3Sigma-deltaModulatorandCompensator
Thebuildingblocksforthesigma-deltamodulatorcanbeeasilysimulatedbyarelayandasampleandhold.Theprecedingcompensatorisrequiredtostabilizetheloopandensureahigh-frequencylimitcycleintheunforcedconditionbyaddingazeroatafrequencyatapproximately1/10ofthesamplingfrequency.
2.4FeedbackArrangement
Inthefeedbackpath,eachoftheeighttopandbottomelectrodesiseitherenergizedbythefeedbackvoltageorheldatzeropotential,dependingontheoutputstateofthefourcomparators.Thecontrolsystemhastoensurethattheelectrodefurtherawayfromthediskisenergized,consequentlyforcesandmomentsaregeneratedtomovethediskbacktothemid-positionparalleltothetopandbottomelectrodes.Providedthediskisheldatzeropotential(whichisassumedhere)themagnitudeoftheelectrostaticforceinthez-directioncanbecalculatedandisgivenby:
forthetoprightelectrode,whereVfbisthefeedbackvoltagewhichiseitherafixedvoltageorzerodependingonthestateofthecomparatoroutput.Theresultofusingtheequationisillustratedinfig.2b.Asaminimumvaluetheelectrostaticforcemustatleastbeabletocounterbalancethegravitationalforceonthedisk,however,anylargerinertialforceswouldmakethedisktouchtheelectrodes,hencethesystemmomentarilyinoperative.Simpleforcecalculationrevealthatafeedbackvoltageof40Vcancounterbalanceapproximately100ginertialforceforadiskwithathicknessof200mandaradiusof0.5mm,whichisbelievedtobesufficient.Themagnitudeofthemomentgeneratedbythefeedbackvoltagecanbecalculatedandisgivenby:
forthemomentaboutthey-axisforthetoprightelectrode.Theresultofusingtheequationisillustratedinfig.2c.Boththeexpressionsfortheforceandmomentssufferfromthesamesingularityasthecapacitanceequationcausingnumericalproblems.Thesamemethodasdescribedforthecapacitancewaschosentocircumventtheproblem.Sincetheexpressionforthemomentscontainasquared
2.5OverallModel
Fig.3showstheoverallmodelimplementedinSimulink.Thenonlinearexpressionshavebeendirectlyimplementedasfunctionblocks.Onesamplingperiodisdividedinapositionmeasuringphaseandaactuationphase,thedelaybetweenandthelengthareimportantparametersdeterminingtheloopstabilitysincetheyaddfurtherlag.
3RESULTS
Simulationswithinitialconditionsassumingthediskalreadyatthemid-positionshowtheexpectedbehaviour;thefouroutputsignalsfromthecomparatorsarerepetitivesequencesoftwohighperiodsandtwol
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