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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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