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ALEA
8DICEMBRE201513DICEMBRE2015 ENRICOS.
NTCMeasurements
Whendealingwithlowcost,homemadetermometers,thechoiceoftenfallontheNTCs.Thesesensorsareresistorswhicharevaryingtheirresistanceinfunctionofthetemperature,withanegativetrend:NegativeTemperatureCoefficientistheirname,alsoknownasThermistor.Thatmeanshigherthetemperature,lowertheresistance:
AtypicalNTCcurve(www.arroyinstruments.com)
Weobservethatthischeapsensorhavethepricetobenonlinearatall.Therearevariousmethodstoderiveamethodtointerpretthecorrecttemperature.HereIwillgothroughtheonesthatareusedtoachievequitereasonableprecisionwithouthaving/payingacalibrationlaboratory.
LookUpTableapproach
Usuallythedatasheetprovidesasetofvalueswhicharesampledfromasensorandcorrespondtoourbeinginsidetheirstatedtolerance:
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Inthispictureyoucanseethestatedresistanceatacertaintemperature,anditstolerance.Animmediatedrawback,ifnoothertablesareavailable,isthatthistablecanbringaprecisionofupto5Conly.TheparameterBiscallednormally .
Moreover,asensorwithatoleranceof5%,refersto25Conly.Thiserroroftheresistancemustbeaddedwiththe duetotoll.factorinthetableabove,fromnowoncalled/.Associatedwithit,thereisthetemperaturecoefficient,TCR(alsoknownas ),whichdescribeshowsteepisthecurve.Youmayunderstandthatwithaverysteepcurve,soanhighTCR,thereisalittledeviationofthetemperatureintheXaxis,associatedwithahighvariationofresistanceontheYaxisthereforewehaveanhighsensitivity.ThesituationistheoppositewhenreadinghightemperatureswithalowTCR,thesensitivitydropsrapidly.
Thethermistorerrorofapunctualreadisthendevisedasfollow:
(eq.1)
where istheresistancetoleranceatthereferencetemperature(specifiedinthedatasheet,hereis25C),and isacoefficientwhichcharacterizetheNTCmaterial,devisedbymeasuringtwodifferenttemperature(andisspecifiedinthedatasheet).Ifweneedtokeeptherighttoleranceafterchangingthesensorwithanotherofthesamemodelwithoutcalibration,weneedtoobservethesetolerances.CombineingthemwiththeTCR,weobtainthetemperaturespunctualerror:
(eq.2)
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(eq.2)
Equationapproach
Therearealotofmethodswhichareusedtolinearizethebehaviour,bylinearizingthemodel.ThemostfamousistheSteinhartHartequation,whichusesasetofcoefficientswhichareprovidedintwoways:frommanufacturer,orcanbedevisedbymeasuring3differenttemperaturesandsolving3equationsfor3unknowns.Ifthesecoeffientsarenotprovidedfrommanufacturer,onecanuseamoreprecisetermometer,measure3temperaturesanditsresistancetemperature,solvingthisequationfortheSHof3rdorder:
Thesensorsmanufactureradoptedtoexperimentprovidesthecoefficientsupto4thorder,bothforreversedanddirectmeasurement:
Withasensorstatedtohave5%oftolerance,onecanactuallyusethecoefficientwithoutthefulldecimalprecisioninsteadused(ideally)fromthemanufacturer,becausetheerrorprovidedbysuchformulaisintheorderofmK,whilethefinalreading,duetovariouserrors,ishigherthan1.5K.IneglectthisSHerror.
Errorestimation
AssuminganegligibleerrorfromtheSHcalculusthatusestheparametersgivenby
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AssuminganegligibleerrorfromtheSHcalculusthatusestheparametersgivenbymanufacturer,usuallyweuseamicrocontrollerwithanADCandavoltagedivider.TheerrorfromacommonADCishalfLSB,sothat
(eq.3)
Theerrorofthevoltagedividertendstobedoubleofthetworesistorsusedifaretoomuchdifferent,otherwisetendstobethemeanofthetworelativeerrorsofthetworesistors,like
.Seethegraph,where100%isthemeanvaluebetweenerrorsofthe2resistors,and200%representthesumofthe2errors:
(electronicproducts.com)
Thesetworesistorsareusedinthisway,inwhichoneistheThermistor:
whereatthereferencetemperature(providedbymanufacturerandusually25C)theRrefandRthermhavethesamevalue(soweboughtamatchedthermistorwithacertainresistancetoreduceerrors).Dependingonyourtemperaturerange,youcansehowvariesthetheratioandseehowgreatercanbeconsideredthetotalerror.
Howgreateristheerrorinmytemperaturerange?Thepreviousvalueof anditsTCRwillleadtoatemperatureerrorlikethis:
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inwhichtheADCandconditioncircuitryerrorsareNOTconsidered.Butanideaofperformancecanbemadeifnocalibrationisperformed(seelater).Iftheresistanceattheextremesofmyrangeisnotsodifferentfromthereferenceresistor(anormalfixedresistorintheschematicabove),thenitisnotamandatorytosumitupbotherrorsofthetworesistors,butcanbealittleless.
IfIneedtomeasurebetween0Cand100C,thedatasheetprovidestheadditionalerroroftheresistanceduetothetoleranceofparameter,called/.Wewillfindthatat100CthereislowTCRandhighrelativeerror.ONLYNOWwecanapplytheworstcasetotalerrorwith ,where isthe(eq.1)attemperatureof100Cusingthetablefrommanufacturer,whileRrefisthethefixedresistorintheschematicaboveand itsrelativeerror.
TCRwillbechosentoachievethehigherrelativeerror,sowillbetheTCRat100C(assaidbefore,thehighertemperatureoftherange),alongwiththeestimatedresistancevalueatthattemperature(ofcourse..).YoumayseehowtheerrorcanbegreatlyreducedifreadingvalueswithhigherTCRatlowertemperatures,andhowsmallisitat25C.Buttheboundariesmustcontainsthegreatererrortolerance,allowingtheusertochangethesensorinthefieldwithoutrecalibration.
Withthecalibrationusingthesetof3equationsabove,alltheseerrorsarecompensated,voltagedividerincluded.TheremainingonewillbetruncationerroroftheSHcoefficientsduetothefinitemachineprecision(whetherisaPCoranMCUusedtomakethecalibration),theerrorsofthereferencetermometerandtheintrinsicerrorsoftheSHmodel,thequantizationerroroftheADC(halfLSB)andforsureothersthatIhavemissed.Itisnottrivialtoquantifyeverything.Andquantification,whentalkingaboutmeasures,isalmosteverything.
WhereistheADC?
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Wehavefoundhowgreateristheerroroftheanalogquantities.NowwhereistheleastsensiblepartoftheNTCcurve?Theoneatthehighertemperature,assaidbefore(lowerTCR).UntilnowIhaveestimatedacertainerrorofthetotalvoltagedividersresistance.
NowisneededtofindhowanADCerrorcanmismatchtheresistance.Letsgoat100C,usinganSHestimationortheLookUptable,thencalculateasortofmanualderivative,letssaythevalue
,whereRistheimmediateavailablesteptoachieveatemperaturelowerofastepequaltotherequiredprecision(ifIwantaprecisionof1C,thenistheresistanceat99CifIhavetheSHequation,oritistheresistanceat95CifIhavearoughLookUptableliketheoneinthisarticle).
Fromthecircuitofthevoltagedivider,wehave and.The ishowmanydiscretestepscanbesampledinside
aRvariation.E.g.,if1LSB=3mV(ADCprovides3mV/bit)andfrom100Cto99CthevariationreadfromADCis6mV,Icanthaveanaccuracyhigherthan2LSB,meaning0.5C(2LSBtorepresent1C).IfIamlowerthan1LSB,Icantdiscernmyprefixedsteplowerthan1C.
Sayingthesamemoremathematically:considerthereferencevoltageappliedtovoltagedividertobe3V.AndthedatasheetprovidesacertainTCRat100C.Thentheresistanceat99Cwillbe:
sothat:
Finally,theadditionaltemperatureerrorfromtheADCis,intheworstcase:
(eq.4)
Conclusions
Thefinalprecision,from(eq.2)and(eq.4),is:
OnecantryandfindoutthatwithanADCofN=10bit,andcomponentsat5%,includedtheNTC,inrangebetween0C100C,hardlycanbeachievedaprecisionlowerthan3C/4C,despitetheaccuracycanbearound0.5C/1C,withoutcalibration.Butnotethatthislowprecisionisduetotheconsiderationoftherangeuptoitsmostimpreciseextreme:reducingtherangeto,letssay,60Ctheprecisioncanbeimprovedalot.Justkeepthatinmindwhenyouread25C,or150CusinganNTC.
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