(2)
withsthewidthofthegap.Thisdeviationisnegligiblecomparedto(l),whenthegapwidthislessthan1/3ofthedistancebetweentheelectrodes.
Anothercauseoferrorsoriginatesfromapossiblefiniteskewangleαbetweenthetwoelectrodes(Fig.3).Assumingthefollowingconditions:
thepotentialsonthesmallelectrodeandtheguardelectrodeareequalto0V,
thepotentialonthelargeelectrodeisequaltoVvolt,
theguardelectrodeislargeenough,
itcanbeseenthattheelectricfieldwillbeconcentric.
Fig.3.Electrodeswithangleα.
Tokeepthecalculationssimple,wewillassumetheelectrodestobeinfinitelylargeinonedirection.Nowtheproblemisatwo-dimensionalonethatcanbesolvedbyusingpolar-coordinates(r,φ).Inthiscasetheelectricalfieldcanbedescribedby:
(3)
Tocalculatethechargeonthesmallelectrode,wesetφto0andintegrateoverr:
(4)
withBltheleftborderofthesmallelectrode:
(5)
andBrtherightborder:
(6)
Solving(4)resultsin:
(7)
Forsmallα'sthiscanbeapproximatedby:
(8)
Itappearstobedesirabletochooselsmallerthand,sotheerrorwilldependonlyontheangleα.Inourcase,achangeintheangleof0.6°willcauseanerrorlessthan100ppm.
Withaproperdesigntheparametersεoandlareconstant,andthenthecapacitancebetweenthetwoelectrodeswilldependonlyonthedistancedbetweentheelectrodes.
Ⅲ.ELIMINATIONOFPARASITICCAPACITANCES
BesidesthedesiredsensorcapacitanceC,therearealsomanyparasiticcapacitancesintheactualstructure(Fig.2).ThesecapacitancescanbemodeledasshowninFig.4.HereCplrepresentstheparasiticcapacitancesfromtheelectrodeE1andCp2fromtheelectrodeE2totheguardelectrodesandtheshielding.ParasiticcapacitanceCp3resultsfromimperfectshieldingandformsanoffsetcapacitance.WhenthetransducercapacitanceCxisconnectedtoanACvoltagesourceandthecurrentthroughtheelectrodeismeasured,CplandCp2willbeeliminated.Cp3canbeeliminatedbyperforminganoffsetmeasurement.
Fig.4.Eliminationofparasiticcapacitances
Thecurrentismeasuredbytheamplifierwithshuntfeedback,whichhasaverylowinputimpedance.Toobtaintherequiredlinearity,theunity-gainbandwidthfToftheamplifierhastosatisfythefollowingcondition:
(9)
whereTistheperiodoftheinputsignal.
SinceCp2consistsofcablecapacitancesandtheinputcapacitanceoftheopamp,itmayindeedbelargerthanCfandcannotbeneglected.
IV.THECONCEPTOFTHESYSTEM
Thesystemusesthethree-signalconceptpresentedin[2],whichisbasedonthefollowingprinciples.WhenwemeasureacapacitorCxwithalinearsystem,weobtainavalue:
(10)
wheremistheunknowngainandMoff,theunknownoffset.ByperformingthemeasurementofareferencequantityCref,inanidenticalwayandbymeasuringtheoffset,Moff,bymakingm=0,theparametersmandMoffareeliminated.ThefinalmeasurementresultPisdefinedas:
(11)
Inourcase,forthesensorcapacitanceC,itholdsthat:
(12)
whereAxistheareaoftheelectrode,doistheinitialdistancebetweenthem,εisthedielectricconstantand△disthedisplacementtobemeasured.Forthereferenceelectrodesitholdsthat:
(13)
withAreftheareaanddrefthedistance.Substitutionof(12)and(13)into(10)andtheninto(11)yields:
(14)
Here,Pisavaluerepresentingthepositionwhilea1anda0areunknown,butstableconstants.Theconstanta1=Aref/Axisastableconstantprovidedthereisagoodmechanicalmatchingbetweentheelectrodeareas.Theconstantao=(Arefd0/(Axdref)willalsobeastableconstantprovidedthatdoanddrefareconstant.Theseconstantscanbedeterminedbyaone-timecalibration.Inmanyapplicationsthiscalibrationcanbeomitted;whenthedisplacementsensorispartofalargersystem,anoverallcalibrationisrequiredanyway.Thisoverallcalibrationeliminatestherequirementforaseparatedeterminationofa1anda0.
V.THECAPACITANCE-TO-PERIODCONVERSION
Thesignalswhichareproportionaltothecapacitorvaluesareconvertedintoaperiod,usingamodifiedMartinoscillator[4](Fig.5j.
WhenthevoltageswingacrossthecapacitorisequaltothatacrosstheresistorandtheNANDgatesareswitchedoff,thisoscillatorhasaperiodToff:
Toff=4RCoff.(15)
Sincethevalueoftheresistoriskeptconstant,theperiodvariesonlywiththecapacitorvalue.Now,byswitchingontherightNANDport,thecapacitanceCXcanbeconnectedinparalleltoCoff.Thentheperiodbecomes:
Tx=4R(Coff+Cx)=4RCx+Toff(16)
TheconstantsRandToffareeliminatedinthewaydescribedinSectionIV.
In[2]itisshownthatthesystemisimmuneformostofthenonidealitiesoftheopampandthecomparator,likeslewing,limitationsofbandwidthandgain,offsetvoltages,andinputbiascurrents.Thesenonidealitiesonlycauseadditiveormultiplicativeerrorswhichareeliminatedbythethree-signalapproach.
VI.PERIODMEASUREMENTWITHAMICROCONTROLLER
Performingperiodmeasurementwithamicrocontrollerisaneasytask.Inourcase,anINTEL87C51FAisused,whichhas8kByteROM,256ByteRAM,andUARTforserialcommunication,andthecapabilitytomeasureperiodswitha333nsresolution.Eventhoughthecountersare16bwide,theycaneasilybeextendedinthesoftwareto24bormore.
Theperiodmeasurementtakesplacemostlyinthehardwareofthemicrocontroller.Therefore,itispossibletolettheCPUofthemicrocontrollerperformothertasksatthesametime(Fig.6).Forinstance,simultaneouslywiththemeasurementofperiodTx,periodTrefandperiodToff,therelativecapacitancewithrespecttoCrefiscalculatedaccordingto(11),andtheresultistransferredthroughtheUARTtoapersonalcomputer.
Fig.5.ModifiedMartinoscillatorwithmicrocontrollerandelectrodes.
Fig.6.Periodmeasurementasbackgroundprocess.
Fig.7.Positionerrorasfunctionofthepositionandestimateofthenonlinearity.
VII.EXPERIMENTALRESULTS
Thesensorisnotsensitivetofabricationtolerancesoftheelectrodes.Thereforeinourexperimentalsetupweusedsimpleprintedcircuitboardtechnologytofabricatetheelectrodes,whichhaveaneffectiveareaof12mm×12mm.Theguardelectrodehasawidthof15mm,whilethedistancebetweentheelectrodesisabout5mm.Whenthedistancebetweentheelectrodesisvariedovera1mmrange,thecapacitancechangesfrom0.25pFto0.3pF.Thankstothechosenconcept,evenasimpledualopamp(TLC272AC)andCMOSNAND’scouldbeused,allowingasingle5Vsupplyvoltage.Thetotalmeasurementtimeamountstoonly100ms,wheretheoscillatorwasrunningatabout10kHz.
Thesystemwastestedinafullyautomatedsetup,usinganelectricalXYtable,thedescribedsensorandapersonalcomputer.Toachievetherequiredmeasurementaccuracythesetupwasautozeroedeveryminute.Inthiswaythenonlinearity,long-termstabilityandrepeatabilityhavebeenfoundtobetterthan1μmoverarangeof1mm(Fig.7)