3198.4ElectricalResistanceThermometryByproperlyconstructinganRTD,andcorrectlymeasuring its resistance,anuncertaintyintempera-turemeasurementaslowas±0.005°Cispossible.Becauseofthispotentialforlowuncertaintiesandthepredictable and stablebehaviorofplatinum, theplatinumRTDis widely used as alocal standard.For an NIST-certified RTD, a table and interpolating equation would be provided.ResistanceTemperatureDeviceResistanceMeasurementTheresistance of anRTDmaybemeasured bya number of means,and thechoiceof anappropriate resistance measuring device mustbe made based on the required level of uncertaintyinthefinal temperaturemeasurement.Conventional ohmmeters cause a small currenttoflowduringresistancemeasurements.creatingself-heating intheRTD.Anappreciabletemperaturechange of the sensor maybe caused bythis current,in effect causing aloading error.This is animportantconsiderationforRTDsBridgecircuits,asdescribedin Chapter6,are usedtomeasuretheresistanceof RTDstominimizeloading errors andprovidelowuncertaintiesin measuredresistancevalues.Wheatstonebridge circuits are commonly used for these measurements.However, the basic Wheatstone bridgecircuit does not compensate for the resistance of the leads in measuring the resistance of an RTD,which is a major source of error in electrical resistance thermometers.When greater accuracies arerequired, three-wire and four-wire bridge circuits are used. Figure 8.7a shows a three-wireCallendar-Griffiths bridge circuit.The lead wires numbered 1,2,and 3 have resistances'i,'2, and r3, respectively. At balanced conditions neglecting lead wire effects,R1R3(8.6)R2RRTDbut with the lead wireresistances included inthecircuit analysis,R1R3+r1(8.7)R2=RRTD + r3RR.R3LeadLeadLead7wireswireswiresRTDRTDRTDRRTDRRTDRRTD(a)(b)(c)Figure 8.7 Bridge circuits. (a) Callender-Griffiths 3-wire bridge; (b) and (c) Mueller 4-wire bridge. An averageof the readings in (b)and (c)eliminates the effect of lead wire resistances
E1C08 09/14/2010 14:53:56 Page 319 By properly constructing an RTD, and correctly measuring its resistance, an uncertainty in temperature measurement as low as 0.005C is possible. Because of this potential for low uncertainties and the predictable and stable behavior of platinum, the platinum RTD is widely used as a local standard. For an NIST-certified RTD, a table and interpolating equation would be provided. Resistance Temperature Device Resistance Measurement The resistance of an RTD may be measured by a number of means, and the choice of an appropriate resistance measuring device must be made based on the required level of uncertainty in the final temperature measurement. Conventional ohmmeters cause a small current to flow during resistance measurements, creating self-heating in the RTD. An appreciable temperature change of the sensor may be caused by this current, in effect causing a loading error. This is an important consideration for RTDs. Bridge circuits, as described in Chapter 6, are used to measure the resistance of RTDs to minimize loading errors and provide low uncertainties in measured resistance values. Wheatstone bridge circuits are commonly used for these measurements. However, the basic Wheatstone bridge circuit does not compensate for the resistance of the leads in measuring the resistance of an RTD, which is a major source of error in electrical resistance thermometers. When greater accuracies are required, three-wire and four-wire bridge circuits are used. Figure 8.7a shows a three-wire Callendar-Griffiths bridge circuit. The lead wires numbered 1, 2, and 3 have resistances r1;r2; and r3, respectively. At balanced conditions neglecting lead wire effects, R1 R2 ¼ R3 RRTD ð8:6Þ but with the lead wire resistances included in the circuit analysis, R1 R2 ¼ R3 þ r1 RRTD þ r3 ð8:7Þ Ei RRTD RTD Lead wires (a) r1 R3 R1 R2 r2 r3 G Ei RRTD RTD Lead wires (b) r1 R3 R1 R2 r2 r3 r4 r2 G Ei RRTD RTD (c) r3 R3 ' R1 R2 r4 r1 G Lead wires Figure 8.7 Bridge circuits. (a) Callender-Griffiths 3-wire bridge; (b) and (c) Mueller 4-wire bridge. An average of the readings in (b) and (c) eliminates the effect of lead wire resistances. 8.4 Electrical Resistance Thermometry 319
320Chapter8TemperatureMeasurementsand with R = R2, the resistance of the RTD, RrTD, can be found as(8.8)RRTD = R3 + r1- r3If ri = r3, the effect of these lead wires is eliminated from the determination of the RTD resistanceby this bridge circuit. Note that the resistance of lead wire 2does not contribute to any error in themeasurementatbalancedconditions.sincenocurrentflowsthroughthegalvanometerGThe four-wire Mueller bridge, as shown in Figure 8.7b,c, provides increased compensation forlead-wireresistancescomparedtotheCallendar-Griffithsbridgeand isusedwithfour-wireRTDs.The four-wire Mueller bridge is typically used when low uncertainties are desired, as in cases wheretheRTD is used as a laboratory standard. A circuit analysis of the bridge circuit in the firstmeasurement configuration, Figure 8.7b, yields(8.9)RRTD +r3= R3 +riand in the second measurement configuration,Figure8.7c,yieldsRRTD+r=R',+ r3(8.10)whereR,and Rsrepresent theindicated values of resistanceinthefirst and second configurations.respectively.AddingEquations8.8and8.9resultsinanexpressionfortheresistanceoftheRTDintermsoftheindicatedvaluesforthetwomeasurements:R3 +R3(8.11)RRTD2With this approach, the effect of variations in lead wire resistances is essentially eliminated.Example 8.1An RTD forms one arm of an equal-arm Wheatstone bridge,as shown inFigure8.8.Thefixedresistances,Rand Rgareequal to252.TheRTDhas a resistanceof 25ata temperature of 0°Cand is used to measure a temperature that is steady in time.The resistance of the RTD over a small temperature rangemay be expressed,as in Equation 8.5:RRTD = Ro[1 + α(T - To)]PRFigure8.8RTDWheatstonebridgearrangement
E1C08 09/14/2010 14:53:56 Page 320 and with R1 ¼ R2, the resistance of the RTD, RRTD, can be found as RRTD ¼ R3 þ r1 r3 ð8:8Þ If r1 ¼ r3, the effect of these lead wires is eliminated from the determination of the RTD resistance by this bridge circuit. Note that the resistance of lead wire 2 does not contribute to any error in the measurement at balanced conditions, since no current flows through the galvanometer G. The four-wire Mueller bridge, as shown in Figure 8.7b,c, provides increased compensation for lead-wire resistances compared to the Callendar-Griffiths bridge and is used with four-wire RTDs. The four-wire Mueller bridge is typically used when low uncertainties are desired, as in cases where the RTD is used as a laboratory standard. A circuit analysis of the bridge circuit in the first measurement configuration, Figure 8.7b, yields RRTD þ r3 ¼ R3 þ r1 ð8:9Þ and in the second measurement configuration, Figure 8.7c, yields RRTD þ r1 ¼ R 0 3 þ r3 ð8:10Þ where R3 and R 0 3 represent the indicated values of resistance in the first and second configurations, respectively. Adding Equations 8.8 and 8.9 results in an expression for the resistance of the RTD in terms of the indicated values for the two measurements: RRTD ¼ R3 þ R 0 3 2 ð8:11Þ With this approach, the effect of variations in lead wire resistances is essentially eliminated. Example 8.1 An RTD forms one arm of an equal-arm Wheatstone bridge, as shown in Figure 8.8. The fixed resistances, R2 and R3 are equal to 25 V. The RTD has a resistance of 25 V at a temperature of 0C and is used to measure a temperature that is steady in time. The resistance of the RTD over a small temperature range may be expressed, as in Equation 8.5: RRTD ¼ R0½ 1 þ að Þ T T0 R2 R1 R3 Ei G RTD RRTD Figure 8.8 RTD Wheatstone bridge arrangement. 320 Chapter 8 Temperature Measurements
8.4ElectricalResistanceThermometry321Suppose the coefficient of resistance for this RTD is 0.003925°C-1.A temperature measurementis made by placing the RTD in the measuring environment and balancing the bridge byadjustingR.The value of R, required to balance the bridge is 37.362.Determine the temperature of the RTD.KNOWNR(0°C)=250α=0.003925°C-1Ri=37.36 (whenbridge isbalanced)FIND Thetemperatureof theRTDSOLUTION The resistance of the RTD is measured bybalancing the bridge; recall that in abalanced conditionR3RRTD = R1R2The resistanceof theRTDismeasuredtobe37.362.With Ro=25atT=0°Candα=0.003925°C-1,Equation8.5becomes37.36Q=25(1+αT)0The temperature of the RTD is 126°C.Example8.2Consider the bridge circuit and RTD of Example 8.1. To select or design a bridge circuit formeasuring the resistance of the RTD in this example,the required uncertainty in temperature wouldbe specified. If the required uncertainty in the measured temperature is ≤ 0.5°C, would a 1% totaluncertainty in each of the resistors that make up the bridge be acceptable? Neglect the effects of leadwire resistances for this example.KNOWNA required uncertainty intemperatureof±0.5°C,measured withtheRTDand bridgecircuitofExample8.1.FINDThe uncertaintyin the measured temperature for a 1% total uncertainty in each of theresistors that makeup the bridgecircuit.ASSUMPTIONAll uncertainties are provided and evaluated at the 95% confidence level.SOLUTIONPerform a design-stage uncertainty analysis.Assuming at thedesign stage that thetotal uncertainty in the resistances is 1%, then with initial values of the resistances in the bridgeequal to 25, the design-stage uncertainties are set atUR:=UR2=UR,=(0.01)(25)=0.25The root-sum-squares (RsS)method is used to estimate the propagation of uncertainty in eachresistor to the uncertainty in determiningtheRTDresistancebyTOR[ORORURTDURR[aRiaR2OR
E1C08 09/14/2010 14:53:56 Page 321 Suppose the coefficient of resistance for this RTD is 0.003925C1 . A temperature measurement is made by placing the RTD in the measuring environment and balancing the bridge by adjusting R1. The value of R1 required to balance the bridge is 37:36 V. Determine the temperature of the RTD. KNOWN Rð0CÞ ¼ 25 V a ¼ 0:003925C1 R1 ¼ 37:36 V (when bridge is balanced) FIND The temperature of the RTD. SOLUTION The resistance of the RTD is measured by balancing the bridge; recall that in a balanced condition RRTD ¼ R1 R3 R2 The resistance of the RTD is measured to be 37:36 V. With R0 ¼ 25 V at T ¼ 0C and a ¼ 0:003925 C1, Equation 8.5 becomes 37:36 V ¼ 25 1ð Þ þ aT V The temperature of the RTD is 126C. Example 8.2 Consider the bridge circuit and RTD of Example 8.1. To select or design a bridge circuit for measuring the resistance of the RTD in this example, the required uncertainty in temperature would be specified. If the required uncertainty in the measured temperature is 0:5 C, would a 1% total uncertainty in each of the resistors that make up the bridge be acceptable? Neglect the effects of lead wire resistances for this example. KNOWN A required uncertainty in temperature of 0.5C, measured with the RTD and bridge circuit of Example 8.1. FIND The uncertainty in the measured temperature for a 1% total uncertainty in each of the resistors that make up the bridge circuit. ASSUMPTION All uncertainties are provided and evaluated at the 95% confidence level. SOLUTION Perform a design-stage uncertainty analysis. Assuming at the design stage that the total uncertainty in the resistances is 1%, then with initial values of the resistances in the bridge equal to 25 V, the design-stage uncertainties are set at uR1 ¼ uR2 ¼ uR3 ¼ ð Þ 0:01 ð Þ¼ 25 0:25 V The root-sum-squares (RSS) method is used to estimate the propagation of uncertainty in each resistor to the uncertainty in determining the RTD resistance by uRTD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qR qR1 uR1 2 þ qR qR2 uR2 2 qR qR3 uR3 2 s 8.4 Electrical Resistance Thermometry 321
322Chapter8Temperature MeasurementswhereR;R3R = RRTDR2and we assume that the uncertainties are not correlated. Then, the design-stage uncertainty in theresistanceof theRTDisR3R;R3R1URTIRR2R2/(1×0.25)2+ (1×0.25)2+(1×0.25)2=0.433 URTD=To determine the uncertainty in temperature, we knowR = RRTD = Ro[1 + α(T - To)]andOTurURTDORSettingTo=0CwithRo=252,andneglectinguncertaintiesinToα,andRo.wehaveOT1aRaRo11αRo(0.003925°C-1)(25 2)Then the design-stage uncertainty in temperature is(OT)0.433=4.4℃UT =URTDOR0.0980/°cThe desired uncertainty in temperature is not achieved with the specified levels of uncertainty in thepertinent variables.COMMENTUncertainty analysis,in this case,would haveprevented performing a measure-ment that would not provideacceptableresults.Example8.3Suppose the total uncertaintyin thebridgeresistances of Example8.1 was reduced to0.1%.Wouldthe required level of uncertainty intemperaturebeachieved?KNowNThe uncertainty in each of theresistors in the bridge circuitfor temperaturemeasurement from Example 8.1is±0.1%.FIND The resulting uncertainty in temperature.SOLUTION The uncertainty analysis from the previous example may bedirectly applied, withthe uncertainty values for the resistances appropriately reduced.The uncertainties for theresistances
E1C08 09/14/2010 14:53:57 Page 322 where R ¼ RRTD ¼ R1R3 R2 and we assume that the uncertainties are not correlated. Then, the design-stage uncertainty in the resistance of the RTD is uRTD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R3 R2 uR1 2 þ R1R3 R2 2 uR2 2 þ R1 R2 uR3 2 s uRTD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 0:25 2 þ ð Þ 1 0:25 2 þ ð Þ 1 0:25 2 q ¼ 0:433 V To determine the uncertainty in temperature, we know R ¼ RRTD ¼ R0½ 1 þ að Þ T T0 and uT ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qT qR uRTD 2 s Setting T0 ¼ 0C with R0 ¼ 25 V, and neglecting uncertainties in T0, a, and R0, we have qT qR ¼ 1 aR0 1 aR0 ¼ 1 0:003925 C1 ð Þ 25 V Then the design-stage uncertainty in temperature is uT ¼ uRTD qT qR ¼ 0:433 V 0:098 V= C ¼ 4:4 C The desired uncertainty in temperature is not achieved with the specified levels of uncertainty in the pertinent variables. COMMENT Uncertainty analysis, in this case, would have prevented performing a measurement that would not provide acceptable results. Example 8.3 Suppose the total uncertainty in the bridge resistances of Example 8.1 was reduced to 0.1%. Would the required level of uncertainty in temperature be achieved? KNOWN The uncertainty in each of the resistors in the bridge circuit for temperature measurement from Example 8.1 is 0.1%. FIND The resulting uncertainty in temperature. SOLUTION The uncertainty analysis from the previous example may be directly applied, with the uncertainty values for the resistances appropriately reduced. The uncertainties for the resistances 322 Chapter 8 Temperature Measurements
3238.4ElectricalResistanceThermometryare reduced from 0.25 to 0.025, yielding(1×0.025)2+(1×-0.025)+(1×0.025)2=0.04330URTD =and the resulting 95% uncertainty interval in temperature is ±0.44°C, which satisfies the designconstraint.COMMENT This result provides confidence that the effect of the resistors'uncertainties will notcause the uncertainty in temperatureto exceed the target value.However,the uncertainty in temperaturealso depends on other aspects of the measurement system. The design-stage uncertainty analysisperformed in this example may be viewed as ensuring that the factors considered do not produce ahigher than acceptable uncertainty level.Practical ConsiderationsThetransientthermal responseoftypicalcommercialRTDs isgenerallyquiteslowcomparedwithothertemperature sensors, and for transient measurements bridge circuits must be operated in a deflectionmode or use expensive auto-balancing circuits.For these reasons, RTDs are not generally chosen fortransient temperature measurements.A notable exception is the use of very small platinum wires andfilms for temperature measurements in noncorrosive flowing gases. In this application, wires havingdiameters on theorderof 10μm can havefrequency responses higher than any othertemperature sensor,because of their extremely low thermal capacitance.Obviously,the smallest impact would destroythissensor.Otherresistance sensorsin the form of thinmetallicfilmsprovidefasttransientresponsetemperature measurements,often in conjunction with anemometry or heat flux measurements. Suchmetallic films are constructed by depositing a film, commonly of platinum, onto a substrate and coatingthe film with a ceramic glass for mechanical protection (7).Typical film thickness rangesfrom 1 to2 μm,with a 10-μum protective coating.Continuous exposure at temperatures of 600°C is possible with thisconstruction. Some practical uses for film sensors include temperature control circuits for heatingsystems and cooking devices and surface temperature monitoring on electronic components subjecttooverheating.Uncertaintylevels range from about±0.1 to2°C.ThermistorsThermistors(fromthermally sensitiveresistors)areceramic-likesemiconductordevices.Themost common thermistors areNTC, and theresistance of these thermistors decreases rapidlywithtemperature,which is in contrastto the small increases ofresistance with temperatureforRTDs.Equation 8.2is too simpleto accuratelydescribe resistancechanges over practical temperatureranges;a more accuratefunctional relationship between resistance and temperaturefor a thermistorisgenerallyassumed tobeof theformR = RoeB(1/T-1/To)(8.12)The parameter β ranges from 3500 to 4600 K, depending on the material, temperature, andindividual construction for each sensor, and therefore must be determined for each thermistor.Figure 8.9shows thevariation of resistance with temperaturefor two common thermistor materials;
E1C08 09/14/2010 14:53:57 Page 323 are reduced from 0.25 to 0.025, yielding uRTD ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ 1 0:025 2 þ ð Þ 1 0:025 2 þ ð Þ 1 0:025 2 q ¼ 0:0433 V and the resulting 95% uncertainty interval in temperature is 0.44C, which satisfies the design constraint. COMMENT This result provides confidence that the effect of the resistors’ uncertainties will not cause the uncertainty in temperature to exceed the target value. However, the uncertainty in temperature also depends on other aspects of the measurement system. The design-stage uncertainty analysis performed in this example may be viewed as ensuring that the factors considered do not produce a higher than acceptable uncertainty level. Practical Considerations The transient thermal response of typical commercial RTDs is generally quite slow compared with other temperature sensors, and for transient measurements bridge circuits must be operated in a deflection mode or use expensive auto-balancing circuits. For these reasons, RTDs are not generally chosen for transient temperature measurements. A notable exception is the use of very small platinum wires and films for temperature measurements in noncorrosive flowing gases. In this application, wires having diameters on the order of 10mm can have frequency responses higher than any other temperature sensor, because of their extremely low thermal capacitance. Obviously, the smallest impact would destroy this sensor. Other resistance sensors in the form of thin metallic films provide fast transient response temperature measurements, often in conjunction with anemometry or heat flux measurements. Such metallic films are constructed by depositing a film, commonly of platinum, onto a substrate and coating the film with a ceramic glass for mechanical protection (7). Typical film thickness ranges from 1 to 2mm, with a 10-mm protective coating. Continuous exposure at temperatures of 600C is possible with this construction. Some practical uses for film sensors include temperature control circuits for heating systems and cooking devices and surface temperature monitoring on electronic components subject to overheating. Uncertainty levels range from about 0.1 to 2C. Thermistors Thermistors (from thermally sensitive resistors) are ceramic-like semiconductor devices. The most common thermistors are NTC, and the resistance of these thermistors decreases rapidly with temperature, which is in contrast to the small increases of resistance with temperature for RTDs. Equation 8.2 is too simple to accurately describe resistance changes over practical temperature ranges; a more accurate functional relationship between resistance and temperature for a thermistor is generally assumed to be of the form R ¼ R0ebð Þ 1=T1=T0 ð8:12Þ The parameter b ranges from 3500 to 4600 K, depending on the material, temperature, and individual construction for each sensor, and therefore must be determined for each thermistor. Figure 8.9 shows the variation of resistance with temperature for two common thermistor materials; 8.4 Electrical Resistance Thermometry 323