324Chapter8TemperatureMeasurementsThermistor Resistance10Y0.1KLLβ=3395K0.01WLLLβ = 3900 K1-1110.001500100150200250300Figure8.9Representativethermistorre-T, temperature [°C]sistance variations with temperature.the ordinate is the ratio of the resistance to the resistance at 25°C.Thermistors exhibit largeresistance changes with temperature in comparison to typical RTDas indicated by comparison ofFigures 8.6 and8.9.Equation 8.12 isnotaccurate over a wide range of temperature,unlessβ is takento be a function of temperature; typically the value ofβ specified by a manufacturerfor a sensor isassumed to be constantover a limited temperaturerange.A simple calibration is possible fordetermining β as a function of temperature, as illustrated in the circuits shown in Figure 8.10.Othercircuits and a more complete discussion of measuringβ may be found in the Electronic IndustriesAssociationstandardThermistorDefinitionsandTestMethods (8).E,E;RZ4TaR,CurrentThermisterVoltage(a)(b)Figure 8.10 Circuits for determining β for thermistors.(a) Voltage divider methodRr=Ri(Ei/E-1).Note:BothR,andE,mustbeknownvalues.ThevalueofR,maybevaried to achieve appropriate values of thermistor current. (b) Volt-ammeter method.Note:Bothcurrentandvoltagearemeasured
E1C08 09/14/2010 14:53:57 Page 324 the ordinate is the ratio of the resistance to the resistance at 25C. Thermistors exhibit large resistance changes with temperature in comparison to typical RTD as indicated by comparison of Figures 8.6 and 8.9. Equation 8.12 is not accurate over a wide range of temperature, unless b is taken to be a function of temperature; typically the value of b specified by a manufacturer for a sensor is assumed to be constant over a limited temperature range. A simple calibration is possible for determining b as a function of temperature, as illustrated in the circuits shown in Figure 8.10. Other circuits and a more complete discussion of measuring b may be found in the Electronic Industries Association standard Thermistor Definitions and Test Methods (8). 0 50 100 150 200 250 300 T, temperature [ C] Thermistor Resistance R(T)/R(25 C) 0.001 0.01 0.1 1 10 = 3395 K = 3900 K Figure 8.9 Representative thermistor resistance variations with temperature. R1 Thermister Voltage Current (a) T (b) E1 E1 Ei Ei RT RT A RV T Figure 8.10 Circuits for determining b for thermistors. (a) Voltage divider method: RT ¼ R1ð Þ Ei=E1 1 . Note: Both R1 and Ei must be known values. The value of R1 may be varied to achieve appropriate values of thermistor current. (b) Volt-ammeter method. Note: Both current and voltage are measured. 324 Chapter 8 Temperature Measurements
8.4ElectricalResistanceThermometry325Thermistors are generally used when high sensitivity, ruggedness, or fast response times arerequired (9).Thermistors are often encapsulated inglass, and thus can be used in corrosiveor abrasiveenvironments.The resistance characteristics of the semiconductor material may change at elevatedtemperatures, and some aging ofa thermistor occurs at temperatures above200C.The high resistanceof a thermistor, compared to that of an RTD, eliminates the problems of lead wire resistancecompensation.A commonly reported specification of a thermistor is the zero-powerresistanceanddissipationconstant.The zero-powerresistance of a thermistoris theresistance value of the thermistorwith noflow of electric current.The zero power resistance should be measured such that a decrease in thecurrent flow to the thermistor results in not more than a 0.1% change in resistance.The dissipationconstant for a thermistor is defined at a given ambient temperature asP(8.13)T-T.where=dissipationconstantP=power supplied to the thermistorT,T=thermistor and ambient temperatures,respectivelyExample8.4The output of a thermistor is highly nonlinear with temperature, and there is often a benefit tolinearizing the output through appropriate circuit, whether active or passive. In this examplewe examine the output of an initially balanced bridge circuit where one of the arms contains athermistor.ConsideraWheatstonebridgeas shown inFigure8.8,but replacetheRTD withathermistor havinga valueof Ro=10,000withβ=3680K.Herewe examinetheoutputof thecircuitovertwotemperatureranges:(a)25-325°C,and(b)25-75°C.KNOWNAWheatstonebridgewhereR2=R3=R4=10,000andwhereRisa thermistor.FIND The output of the bridge circuit as a function of temperature.SOLUTIONThe fundamental relationship between resistances in a Wheatstonebridge and thenormalized output voltage is provided in Equation 6.14:RiR3Eo(6.14)Ei(R+R2R3+R4)Andtheresistanceof thethermistor isR = RaeB(1/T-1/T,)Substituting in Equation 6.14 for R yieldsReB(1/T-1/T.)EoR3E;(R,eB(1/T-1/T。) + R2R3 +R4
E1C08 09/14/2010 14:53:57 Page 325 Thermistors are generally used when high sensitivity, ruggedness, or fast response times are required (9). Thermistors are often encapsulated in glass, and thus can be used in corrosive or abrasive environments. The resistance characteristics of the semiconductor material may change at elevated temperatures, and some aging of a thermistor occurs at temperatures above 200 C. The high resistance of a thermistor, compared to that of an RTD, eliminates the problems of lead wire resistance compensation. A commonly reported specification of a thermistor is the zero-power resistance and dissipation constant. The zero-power resistance of a thermistor is the resistance value of the thermistor with no flow of electric current. The zero power resistance should be measured such that a decrease in the current flow to the thermistor results in not more than a 0.1% change in resistance. The dissipation constant for a thermistor is defined at a given ambient temperature as d ¼ P T T1 ð8:13Þ where d ¼ dissipation constant P ¼ power supplied to the thermistor T, T1 ¼ thermistor and ambient temperatures, respectively Example 8.4 The output of a thermistor is highly nonlinear with temperature, and there is often a benefit to linearizing the output through appropriate circuit, whether active or passive. In this example we examine the output of an initially balanced bridge circuit where one of the arms contains a thermistor. Consider a Wheatstone bridge as shown in Figure 8.8, but replace the RTD with a thermistor having a value of R0 ¼ 10; 000 V with b ¼ 3680 K. Here we examine the output of the circuit over two temperature ranges: (a) 25–325C, and (b) 25–75C. KNOWN AWheatstone bridge where R2 ¼ R3 ¼ R4 ¼ 10; 000 V and where R1 is a thermistor. FIND The output of the bridge circuit as a function of temperature. SOLUTION The fundamental relationship between resistances in a Wheatstone bridge and the normalized output voltage is provided in Equation 6.14: Eo Ei ¼ R1 R1 þ R2 R3 R3 þ R4 ð6:14Þ And the resistance of the thermistor is R ¼ Roebð1=T1=ToÞ Substituting in Equation 6.14 for R1 yields Eo Ei ¼ Roebð Þ 1=T1=To Roebð Þ 1=T1=To þ R2 R3 R3 þ R4 8.4 Electrical Resistance Thermometry 325
326Chapter 8TemperatureMeasurements0.50.450.40.350.3国0.25E0.2 0.150.10.0500100200300Temperature (C)(a)0.40.350.30.25E0.20.150.1 -0.05.0-r20506070803040Temperature (°c)(b)Figure8.11Normalized bridge output voltage as a function of temperature with a thermistor as thetemperaturesensor:(a)25°to325°C,(b)25°to75°CFigure 8.1la is a plot of this function over the range 25° to 325°C. Clearly the sensitivity of thecircuit to changes in temperature greatly diminishes as the temperature increases above 100°C, withan asymptoticvalueof 0.5.Figure8.11b shows thebehaviorover the restricted range25°to75°C; alinear curve fit is also shown for comparison. Over this range of temperature, assuming a linearrelationship between normalized output and temperature would be suitable for manyapplicationsprovided thattheaccompanying linearity errorisacceptable
E1C08 09/14/2010 14:53:57 Page 326 Figure 8.11a is a plot of this function over the range 25 to 325C. Clearly the sensitivity of the circuit to changes in temperature greatly diminishes as the temperature increases above 100C, with an asymptotic value of 0.5. Figure 8.11b shows the behavior over the restricted range 25 to 75C; a linear curve fit is also shown for comparison. Over this range of temperature, assuming a linear relationship between normalized output and temperature would be suitable for many applications provided that the accompanying linearity error is acceptable. (a) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 100 200 300 Temperature (o C) Eo/Ei (b) Eo/Ei 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 20 30 40 50 60 70 80 Temperature (oC) Figure 8.11 Normalized bridge output voltage as a function of temperature with a thermistor as the temperature sensor: (a) 25 to 325C, (b) 25 to 75C. 326 Chapter 8 Temperature Measurements
3278.4ElectricalResistanceThermometryExample 8.5The material constant β is to be determined for a particular thermistor using the circuit shown inFigure 8.10a.The thermistor has aresistance of 60k at 25°C.The reference resistor in the circuit,Ri,hasa resistanceof 130.5k2.The dissipation constantis0.09mW/pC.Thevoltage source usedfor the measurement is constant at 1.564 V.The thermistor is to be used at temperatures rangingfrom 100 to 150°C.Determine the value of β.KNowNThetemperaturerangeofinterestisfrom100°to150°C.Ro=60,000Ω2,To=25CE,=1.564V,8=0.09mW/C,R=130.5kΩFINDThe value of βover the temperaturerange from 100°to150°C.SOLUTION The voltage dropacross the fixed resistor is measuredfor threeknown values ofthermistor temperature.The thermistor temperature is controlled and determined by placing thethermistor in a laboratory oven and measuring the temperature of the oven. For each measuredvoltage across the reference resistor, the thermistor resistance Rr is determined fromRT=RThe results of these measurements are as follows:RTTemperatureR,Voltage(0)(°C)(V)1001.5015477.41251.5312812.91501.5451604.9Equation8.12canbeexpressed intheformof alinearequationasRTIn-(8.14)RoToApplying thisequation to the measured data,with Ro=60,000, the three data points above yieldthe following:In(RT/Ro)(1/T - 1/To) [K-1]β(K)2.3946.75 × 10-43546.78.43 × 10-43.0603629.99.92× 10-43.6213650.2COMMENT These results are for constant β and are based on the behavior described byEquation 8.12, over the temperaturerange from Toto the temperatureT.The significance of themeasured differences in β is examined further
E1C08 09/14/2010 14:53:57 Page 327 Example 8.5 The material constant b is to be determined for a particular thermistor using the circuit shown in Figure 8.10a. The thermistor has a resistance of 60 kV at 25C. The reference resistor in the circuit, R1, has a resistance of 130:5 kV. The dissipation constant d is 0.09 mW/C. The voltage source used for the measurement is constant at 1.564 V. The thermistor is to be used at temperatures ranging from 100 to 150C. Determine the value of b. KNOWN The temperature range of interest is from 100 to 150C. R0 ¼ 60; 000 V; T0 ¼ 25 C Ei ¼ 1:564 V; d ¼ 0:09 mW= C; R1 ¼ 130:5 kV FIND The value of b over the temperature range from 100 to 150C. SOLUTION The voltage drop across the fixed resistor is measured for three known values of thermistor temperature. The thermistor temperature is controlled and determined by placing the thermistor in a laboratory oven and measuring the temperature of the oven. For each measured voltage across the reference resistor, the thermistor resistance RT is determined from RT ¼ R1 Ei E1 1 The results of these measurements are as follows: Temperature R1 Voltage RT ( C) (V) (V) 100 1.501 5477.4 125 1.531 2812.9 150 1.545 1604.9 Equation 8.12 can be expressed in the form of a linear equation as ln RT R0 ¼ b 1 T 1 T0 ð8:14Þ Applying this equation to the measured data, with R0 ¼ 60; 000 V, the three data points above yield the following: lnð Þ RT =R0 ð Þ 1=T 1=T0 K1 b(K) 2.394 6:75 104 3546.7 3.060 8:43 104 3629.9 3.621 9:92 104 3650.2 COMMENT These results are for constant b and are based on the behavior described by Equation 8.12, over the temperature range from T0 to the temperature T. The significance of the measured differences in b is examined further. 8.4 Electrical Resistance Thermometry 327
328Chapter8TemperatureMeasurementsThemeasuredvalues ofβ in Example8.5are different ateach valueof temperature.Ifβweretruly a temperature-independent constant,and these measurements had negligible uncertainty,allthree measurements would yield the same value of β. The variation in β may be due to a physicaleffect of temperature, or may be attributable to the uncertainty in the measured values.Arethemeasured differences significant,and ifso,what valueofβbestrepresents thebehaviorof the thermistor over this temperature range? To perform the necessary uncertainty analysis,additional information must be provided concerning the instruments and procedures used in themeasurement.Example 8.6Perform an uncertainty analysis to determine the uncertainty in each measured value ofβ in Example 8.5,and evaluate a singlebestestimate of βforthistemperaturerange.Themeasurement ofβinvolves themeasurement of voltages,temperatures, and resistances.For temperature there is a random errorassociated withspatial andtemporal variations in the oven temperature with arandom standarduncertaintyof st =0.19°C for 20measurements.In addition, based on a manufacturer's specification, there is aknown measurement systematic uncertaintyfor temperature of 0.36°C (95%) in the thermocoupleThesystematic errors inmeasuring resistance and voltagearenegligible,and estimates of theinstrument repeatability, which are based on the manufacturer's specifications in the measuredvalues and assumed to be at a 95% confidence level, are assigned systematic uncertainties of 1.5%forresistanceand0.002Vforthevoltage.KNowN Standard deviation of the means for oven temperature, S =0.19°C, N=20.Theremaining errors are assigned systematic uncertainties at95% confidenceassuming largedegrees offreedom, such that Bx=2bx:BT=2bT =0.36°CBR/R= (2bR)/R =1.5%BE=2bE=0.002VFINDThe uncertainty inβ at each measured temperature, and a best estimatefor β over themeasuredtemperaturerange.SOLUTION Consider theproblem of providing a singlebest estimate of β.One method ofestimation might be to average the three measured values.This results in a value of 3609 KHowever,sincetherelationshipbetween(Rr/Ro)and(1/T-1/To)isexpectedtobelinear,aleast-squares fit can be performed on the three data points, and include the point (O, O). The resulting valueof β is 3638 K. Is this difference significant, and which value best represents the behavior of thethermistor?To answer these questions,an uncertainty analysis is performed forβ.For each measured value,In(Rr/Ro)β21/T-1/ToUncertainties in voltage,temperature,andresistance arepropagated intotheresulting valueofβforeachmeasurement
E1C08 09/14/2010 14:53:57 Page 328 The measured values of b in Example 8.5 are different at each value of temperature. If b were truly a temperature-independent constant, and these measurements had negligible uncertainty, all three measurements would yield the same value of b. The variation in b may be due to a physical effect of temperature, or may be attributable to the uncertainty in the measured values. Are the measured differences significant, and if so, what value of b best represents the behavior of the thermistor over this temperature range? To perform the necessary uncertainty analysis, additional information must be provided concerning the instruments and procedures used in the measurement. Example 8.6 Perform an uncertainty analysis to determine the uncertainty in each measured value of b in Example 8.5, and evaluate a single best estimate of b for this temperature range. The measurement of b involves the measurement of voltages, temperatures, and resistances. For temperature there is a random error associated with spatial and temporal variations in the oven temperaturewith a random standard uncertainty of sT ¼ 0:19C for 20 measurements. In addition, based on a manufacturer’s specification, there is a known measurement systematic uncertainty for temperature of 0.36C (95%) in the thermocouple. The systematic errors in measuring resistance and voltage are negligible, and estimates of the instrument repeatability, which are based on the manufacturer’s specifications in the measured values and assumed to be at a 95% confidence level, are assigned systematic uncertainties of 1.5% for resistance and 0.002 V for the voltage. KNOWN Standard deviation of the means for oven temperature, sT ¼ 0:19C; N ¼ 20. The remaining errors are assigned systematic uncertainties at 95% confidence assuming large degrees of freedom, such that Bx ¼ 2bx: BT ¼ 2bT ¼ 0:36 C BR=R ¼ ð Þ 2bR =R ¼ 1:5% BE ¼ 2bE ¼ 0:002 V FIND The uncertainty in b at each measured temperature, and a best estimate for b over the measured temperature range. SOLUTION Consider the problem of providing a single best estimate of b. One method of estimation might be to average the three measured values. This results in a value of 3609 K. However, since the relationship between ð Þ RT =R0 and 1ð Þ =T 1=T0 is expected to be linear, a leastsquares fit can be performed on the three data points, and include the point (0, 0). The resulting value of b is 3638 K. Is this difference significant, and which value best represents the behavior of the thermistor? To answer these questions, an uncertainty analysis is performed for b. For each measured value, b ¼ lnð Þ RT =R0 1=T 1=T0 Uncertainties in voltage, temperature, and resistance are propagated into the resulting value of b for each measurement. 328 Chapter 8 Temperature Measurements