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14ReactanceVariationandElectromagneticSensorsSensorsandSignalConditioning2Reactancevariationsinacomponentorcircuitofferalternativemeasurementmethodsfromthoseavailablewithresistivesensors.Manyreactancevariationmeasurementmethodsdonotrequireanyphysicalcontactwiththesystemtobemeasured,orwhentheydo,exertaminimalmechanicalloadingeffect.Theinherentnonlinearityofsomeofthemeasurementmethodsusedinthiskindofsensorsisovercomethroughdifferentialsensors.Ontheotherhand,thesemethodslimitthemaximalfrequencyforthemeasurandbecauseitmustbeatleasttentimeslowerthantheexcitationfrequency,whichmustbeanalternatingvoltageorcurrent.Someelectromagneticsensorsareinfactself-generatingsensors,buttheyarediscussedinthischapterbecauseofthesimilaritybetweentheiroutputsignalandthatofsomevariablereactancesensors.3zVariableCapacitorAcapacitorconsistsoftwoelectricconductorsseparatedbyadielectric(solid,liquid,orgas)oravacuum.4.1CapacitiveSensorsThiscapacitancedependsonthegeometricalarrangementoftheconductorsandonthedielectricmaterialbetweenthem.4Therefore,anymeasurandproducingavariationinεr,A,ordwillresultinachangeinthecapacitanceCandcanbeinprinciplesensedbythatdevice.For(a),wehaveFor(b),wehaveFor(c),wehaveFor(d),13Theuseofavariablecapacitorasasensorhasseverallimitations.–fringeeffectsFigure4.1showshowtoreducefringeeffectswithoutchanginggeometricalrelations.14–Theinsulationbetweenplatesmustbehighandconstant.–CapacitiveInterference–Connectingwires–Temperature15Capacitivesensorsarelinearornonlinear,dependingontheparameterthatchangesandwhetherwemeasurethecapacitiveimpedanceoradmittance.Inaparallelplatecapacitor,forexample,theoutputvoltageislinearwhenwemeasuretheadmittance(proportionaltoC)ifεrorAchange,butitisnonlinearifthemeasurandchangestheseparationbetweenplates.16wherex=z/d.Bytakingthederivativeofthisequationtofindthesensitivity,wehaveHence,thesensorisnonlinear.17Thesensitivityisnow18Anothermethodtoobtainalinearvoltagefromasensorwhenthedistancebetweenplatesvariesistomeasure,insteadofitsadmittance,itsimpedance(Section5.1).Differentialcapacitorsalsoyieldanoutputlinearlydependentonthemeasurand(Section4.1.2).Capacitivesensorshavehighoutputimpedance.1920zDifferentialCapacitorAdifferentialcapacitorconsistsoftwovariablecapacitorssoarrangedthattheyundergothesamechangebutinoppositedirections.Forexample,thearrangementinFigure4.6yieldsFigure4.6Differentialcapacitorbasedonthevariationofthedistancebetweenplates.21TherespectivedropinvoltageacrosseachcapacitorisSubstitutingforthecapacitancestheirvaluesasgivenby(4.15a)and(4.15b)yields22Whenwesubtractbothvoltagesweobtain23Therefore,anappropriateoutputsignalconditioningyieldsalinearoutputthathasanincreasedsensitivitycomparedtoasinglecapacitor.IfthemeasurandchangestheareaofC1andC2instead,measuringthedifferencebetweencapacitancescanalsoyieldaresultproportionaltoz.24zVariablereluctancesensorsThereluctanceofacircuitindicatestheamountofmagneticfluxitlinks,duetoanelectriccurrent.Ifitisacurrentflowingalongthecircuititself,wecallitselfinductanceL.OtherwisewecallitmutualinductanceM.Theinductancecanbeexpressedas4.2InductiveSensors252627Ifthemagneticcircuitincludespathsthroughtheairandpathsthroughaferromagneticmaterialplacedinseries,thenthegeneralequationforthereluctanceis28Therefore,anyvariationinN,u(magneticpermeabilityofthematerialinsideandaroundthecoil)orinthegeometry(lorA)caninprinciplebeappliedtosensing.Nevertheless,mostinductivesensorsarebasedonavariationofreluctanceanditisadisplacementthatmodifiesit,usuallymodifyingl0oru.Thosethatmodifyl0arecalledvariablegapsensors,andthosethatmodifyuarecalledmovingcoresensors.Rcanalsochangebecauseofeddycurrents,asdescribedinnextsection.Figure4.10Differentconfigurationsforvariablereluctancesensors.(a)to(d)relyonavariationinthenumberofcoilturnsThesensorsin(c)and(d)aredifferential.Figure4.10Differentconfigurationsforvariablereluctancesensors.(e)to(g)relyonamagneticcoremovement;(h)and(i)relyonagapvariation.Thesensorsin(g),and(i)aredifferential.31Thepropertiesofvariablereluctancesensorsstronglydependonthekindofcore.Sensorswithanaircore-meaningnomagneticcorematerial-workatfrequencieshigherthanthoseavailablewhenusinganironcore,butinductancevariationsaresmaller.Sensorswithacoreofironorotherferromagneticmaterialshouldworkbelowabout20kHzinordertoavoidincreasedlossesinthecore.Furthermore,uchangeswithcurrentintensity,thuslimitingthermssupplyvoltagetoabout15V.Coilwindingsarecostlyandbulky,therebypreventingminiaturization.Integratedmicrocoilsovercometheselimitations.3233Theapplicationofavariableinductancetosensinghasseverallimitations:•straymagneticfields•Fringingmagneticfields•LandRareinverselyproportional.•theoutputisbidirectional.•AlldevicesbasedonmagneticpropertiesofmaterialsworkonlyattemperaturesbelowtheirrespectiveCurietemperatures.34Advantages:Onemainadvantageofinductivesensorsisthattheyarenotaffectedbytheambienthumidityorbyothercontaminantsthatcanhaveanoticeableinfluenceoncapacitivesensors.Alsotheirmechanicalloadingeffectisverysmall,althoughhigherthanthatofavariablecapacitor.Inaddition,t