PART1PART1 INSTRUMENTATIONINSTRUMENTATION3 3CHAPTER1CHAPTER1 INSTRUMENTSTATICSINSTRUMENTSTATICSJerryLeeHallJerryLeeHall DepartmentofMechanicalEngineeringDepartmentofMechanicalEngineering IowaStateUniversityIowaStateUniversity Ames,IowaAmes,IowaSriramSundararajanSriramSundararajan DepartmentofMechanicalEngineeringDepartmentofMechanicalEngineering IowaStateUniversityIowaStateUniversity Ames,IowaAmes,IowaMahmoodNaimMahmoodNaim UnionCarbideCorporationUnionCarbideCorporation Indianapolis,IndianaIndianapolis,Indiana1TERMINOLOGY31TERMINOLOGY3 1.1TransducerCharacteristics31.2Definitions42STATICCALIBRATION62STATICCALIBRATION6 2.1CalibrationProcess6 2.2FittingEquationstoCalibration Data63STATISTICSINTHE3STATISTICSINTHE MEASUREMENTPROCESS9MEASUREMENTPROCESS9 3.1UnbiasedEstimates9 3.2Sampling9 3.3TypesofErrors103.4PropagationofErroror Uncertainty10 3.5UncertaintyInterval12 3.6AmountofDatatoTake14 3.7GoodnessofFit15 3.8ProbabilityDensityFunctions163.9DeterminationofConfidence Limitson173.10ConfidenceLimitson RegressionLines18 3.11InferenceandComparison22REFERENCES31REFERENCES311TERMINOLOGY1TERMINOLOGY1.1TransducerCharacteristics1.1TransducerCharacteristicsAmeasurementsystemextractsinformationaboutameasurablequantityfromsomemedium ofinterestandcommunicatesthismeasureddatatotheobserver.Themeasurementofany variableisaccomplishedbyaninstrumentationsystemcomposedoftransducers.Eachtrans- ducerisanenergyconversiondeviceandrequiresenergytransferintothedevicebeforethe variableofinterestcanbedetected.TheInstrumentSocietyofAmerica(ISA)definestransducerasadevicethatprovidesusableoutputinresponsetoaspecifiedmeasurand.Themeasurandisaphysicalquantity, propertyorconditionwhichismeasured.Theoutputistheelectricalquantity,produced byatransducer,whichisafunctionoftheappliedmeasurand.1 Itshouldbemadeveryclearthattheactofmeasurementinvolvestransferofenergy betweenthemeasuredmediumandthemeasuringsystemandhencethemeasuredquantityMechanical Engineers Handbook: Instrumentation, Systems, Controls, and MEMS, Volume 2, Third Edition. Edited by Myer Kutz Copyright 2006 by John Wiley theoutputofapressure gagemaybezeroatatmosphericpressure.Zerodriftisthechangeinoutputfromitssetzerovalueoveraspecifiedperiodoftime. Zerodriftoccursduetochangesinambientconditions,changesinelectricalcondi- tions,agingofcomponents,ormechanicaldamage.Theerrorintroducedmaybesignificantwhenatransducerisusedforlong-termmeasurement. Creepisachangeinoutputoccurringoveraspecifictimeperiodwhilethemeasurand isheldconstantatavalueotherthanzeroandallenvironmentalconditionsareheld constant.1Terminology5 5Figure1Figure1Schematicsillustratingconceptsof(a)accuracyandprecision,(b)hysteresis,(c)astaticerror band,and(d)fittingacurvetothecalibrationdata.Accuracyisthemaximumamountofdifferencebetweenameasuredvariableandits truevalue.Itisusuallyexpressedasapercentageoffull-scaleoutput.Inthestrictest sense,accuracyisneverknownbecausethetruevalueisneverreallyknown.Precisionisthedifferencebetweenameasuredvariableandthebestestimate(asob- tainedfromthemeasuredvariable)ofthetruevalueofthemeasuredvariable.Itisa measureofrepeatability.Precisemeasurementshavesmalldispersionbutmayhave pooraccuracyiftheyarenotclosetothetruevalue.Figure1ashowsthedifferences betweenaccuracyandprecision. Linearitydescribesthemaximumdeviationoftheoutputofaninstrumentfromabest-fittingstraightlinethroughthecalibrationdata.Mostinstrumentsaredesignedsothat theoutputisalinearfunctionoftheinput.Linearityisbasedonthetypeofstraightlinefittedtothecalibrationdata.Forexample,least-squareslinearityisreferencedto thatstraightlineforwhichthesumofthesquaresoftheresidualsisminimized.The6 6InstrumentStaticstermresidualreferstothedeviationsofoutputreadingsfromtheircorrespondingvaluesonthestraightlinefittedthroughthedata. Hysteresisisthemaximumdifferenceinoutput,atanymeasuredvaluewithinthespec-ifiedrange,whenthevalueisapproachedfirstwithincreasingandthenwithdecreas- ingmeasurand.Hysteresisistypicallycausedbyalagintheactionofthesensingelementofthetransducer.Loadingtheinstrumentthroughacycleoffirstincreasing values,thendecreasingvalues,ofthemeasurandprovidesahysteresisloop,asshown inFig.1b.Hysteresisisusuallyexpressedinpercentoffull-scaleoutput. Errorbandisthebandofmaximumdeviationofoutputvaluesfromaspecifiedrefer- encelineorcurve.Astaticerrorband(seeFig.1c)isobtainedbystaticcalibration. Itisdeterminedonthebasisofmaximumdeviationsobservedoveratleasttwo consecutivecalibrationcyclessoastoincluderepeatability.Errorbandaccountsfor deviationsthatmaybeduetononlinearity,nonrepeatability,hysteresis,zeroshift, sensitivityshift,andsoforth.Itisaconvenientwaytospecifytransducerbehaviorwhenindividualtypesofdeviationsneednotbespecifiednordetermined.52STATICCALIBRATION2STATICCALIBRATION2.1CalibrationProcess2.1CalibrationProcessCalibrationistheprocessofcomparisonoftheoutputofameasuringsystemtothevalues ofarangeofknowninputs.Forexample,apressuregageiscalibratedbyadevicecalleda dead-weighttester,whereknownpressuresareappliedtothegageandtheou