Thermistor Calibration And The Steinhart-Hart Equation

Thermistor accuracy is primarily a functionof the thermistor calibration and resistancemeasurement accuracy, whereas temperaturestability depends on the controller and controlenvironment.Thermistor Calibration Error in Temperature with LDT-55254OThermistor Calibration Error ( C)32Calibrated5% Tol.1% Tol.1For more information on thermistor selection,see ILX Lightwave Application Note #2,Selecting and Using Thermistors forTemperature Control.0-1-2-20-10010203040506070Temp (O C)Copies of the programs may be obtained fromILX Lightwave free of charge through theDownloads section on www.ilxlightwave.com.Figure 4. Temperature error due to thermistorcalibration error.In ConclusionThermistor calibration, though not difficult,can be time-consuming. Therefore, it isbest to first determine the requirements ofthe application, then pick an appropriatecalibration method. The methods discusses inthis publication are summarized in Table 2.Often, the thermistor manufacturer will provideR-T values, but the accuracy of these valuesdepends on the resistance tolerance of thethermistor. When a high temperature tolerance is required, it is recommended that theseR-T values be discarded and new values bemeasured as described below.The control temperature tolerance willdecrease rapidly if the thermistor is usedoutside of the temperature range in which itwas calibrated. Temperature and resistancevalues should be made at evenly spacedincrements over a range greater than theintended range-of-operation for the thermistor.For some applications, the nominal R-T datais adequate and the Steinhart-Hart constantscan be calculated using “Faster Method 2,”described below.1. Set a nominal temperature and allow it tostabilize.2. Using a precision DMM (accuracy to aminimum of four places) read the resistanceof the reference thermistor and theuncalibrated thermistor.ILX Lightwave Model 520 uncalibratedthermistors are shipped with three-termnominal constant values as follows:C1 1.125C2 2.347C3 0.8553. Read the resistances three times beforechanging to a new nominal temperature.The three readings can be averaged if usingMethod 1, or all readings can be used ifusing Method 2 or Method 3.The two-term nominal constants, for use withthe LDT-5525, are:C1’ 0.99C2’ 2.57Table 2Summary of Calibration Methods*Also used in ILX Lightwave Model 37xx and 39xx Laser Diode Controllers.4. If using Method 1, repeat steps 1-3 for atotal of three temperature settings. Foreither Method 2 or Method 3, repeat themeasurement as many times as practical;these two methods use least-squares fitto determine the constants, and will bemore accurate with a greater number ofmeasurements.Procedure for CalculatingSteinhart-Hart ConstantsTo calculate the constants for a newthermistor, the temperature and resistance ofthat thermistor will need to be measured atseveral different temperatures covering theexpected range of operation.5. The “true” temperature in degrees Celsiuscan be determined using the constantsfor the precision thermistor and using theinverse of Equation 1, shown below asEquation 4.The following procedure requires somemethod to set and control a nominaltemperature and a calibrated precisionthermistor to reference the temperature.(4) T (C1 C2 * ln(R) C3 * ln(R)3) –1 – 273.15For all methods, it is worth noting thatthe ultimate accuracy of the constantsis dependent upon the accuracy of thetemperature and resistance measurements.6If using Method 3, the approximated Steinhart-Hart equation, disregard the term usingthe constant C3 by using Equation 5.(5) T (C1 C2 * ln(R)) –1 – 273.153

(T)-0.0114.99 25.0136.9550.100This data will be used with one of the threemethods listed in Table 1 to determine thethree Steinhart-Hart constants for the newthermistor.Three Methods of Steinhart-HartConstant CalculationFigure 2. Data format for STEIN2.EXE.Note that the temperature and resistance readings must be separated by one space, and thefile terminated with a resistance reading of ‘1.’Method 1, STEIN1.EXESTEIN1.EXE can be run directly fromthe Windows environment, or the Excelspreadsheet can be used (see Appendixfor program listing). The three temperatureand resistance values are entered and theconstant values are returned. If any of theconstant values are negative there is an errorand the data should be checked orre-measured.The data file can be created in Excel and savedas a “.PRN” file to ensure the data is spacedelimited (the program will not function properlyif the data is tab delimited). The constantswill be output in the form required by the ILXLightwave temperature controller.Alternatively, the program called EasySTEIN2.EXE can be used. This program will promptfor the data to be input directly, rather thanusing a separate data file. As with STEIN2.EXE, the constants are output in a formthat is entered directly into the temperaturecontroller. The constant uncertainties are alsocalculated and displayed.The constants are output in the form used byan ILX temperature controller by scaling eachas shown below.(6) C1 C1 * 10(R)3244415534 986459363560-13(7) C2 C2 * 104(8) C3 C3 * 107The program or spreadsheet performs thisscaling so the output values can be entereddirectly into the temperature controller.Both programs use the method described byPhilip R Bevington in “Data Reduction andError Analysis for the Physical Sciences,”McGraw-Hill, New York, 1969. Matrixinversion is used to solve N simultaneousequations, where N is the number of datapairs in the data file (excluding the marker).Coefficients C1, C2, and C3 are determined byMethod 2, STEIN2.EXESTEIN2.EXE uses least-squares-fit errorreduction, so requires a greater number oftemperature/resistance readings to be taken4minimizing 2, the measure of the fit of thecurve to the data.tolerance 10k thermistor is better than 0.6 C at 50 C.Faster Method 2As discussed previously, some manufacturersprovide nominal R-T values with thethermistor. “Nominal” Steinhart-Hartconstant values can be calculated from themanufacturer’s R-T values with Method 2 ifhigh temperature control tolerances are notrequired for a particular application.Method 3, STEIN3.EXESimilar to Method 2, the third method usesleast-squares-fit error reduction to adjustthe R-T curve for a good fit. This method isintended for use when only the first twoSteinhart-Hart constants are used.The R-T data is collected and formatted thesame as described for Method 2, but theprogram titled STEIN3.EXE is used.Temperature error can be calculated usingEquation (4) with the “nominal” constantvalues and the worst-case resistance valuesfrom the tolerance rating. Results of thisexercise are shown in Figure 3 for a 1% and5% tolerance thermistor. The error is theuncertainty in the temperature based on theresistance tolerance when deriving C1, C2,and C3; it does not include any additional errorbased on the uncertainty in the resistancemeasurement.As with Method 2, a faster method usingMethod 3 can be performed by entering thenominal R-T values supplied by the thermistormanufacturer and running STEIN3.EXE. Theresulting “nominal” Steinhart-Hart constantvalues can then be entered into the LDT5525. Temperature error can be calculatedusing Equation 5 with the “nominal”constant values and the worst-case resistancevalues from the tolerance rating.As shown in Figure 3, below 50 C this error isless than about 1.5 C for the 5% tolerance10k thermistor. The error for a typical 1%As shown in Figure 4, below 50 C this erroris less than about 2 C for the 5% tolerance10k thermistor. The error for a typical1% tolerance 10k thermistor is better than 0.8 C at 50 C.Thermistor Calibration Error in Temperature with Eq (4)32Figure 4 also shows the error associatedwith the calibration for a Model 510 (10k )thermistor, used with the Model LDT-5525Temperature Controller when using Equation5. Again, this does not include any error in theresistance measurement.Oto adjust the R-T curve for a good fit. Datashould be entered into an ASCII data file inthe format shown in Figure 2.Thermistor Calibration Error ( C)6. Compile the data into a table with twocolumns: “true” temperature calculatedusing Equation 4, and resistancemeasured from the uncalibrated thermistor.15% Tol.1% Tol.0-1-2-3-20-10010203040506070Temp (O C)Figure 3. Temperature error due to thermistorcalibration error.5

(T)-0.0114.99 25.0136.9550.100This data will be used with one of the threemethods listed in Table 1 to determine thethree Steinhart-Hart constants for the newthermistor.Three Methods of Steinhart-HartConstant CalculationFigure 2. Data format for STEIN2.EXE.Note that the temperature and resistance readings must be separated by one space, and thefile terminated with a resistance reading of ‘1.’Method 1, STEIN1.EXESTEIN1.EXE can be run directly fromthe Windows environment, or the Excelspreadsheet can be used (see Appendixfor program listing). The three temperatureand resistance values are entered and theconstant values are returned. If any of theconstant values are negative there is an errorand the data should be checked orre-measured.The data file can be created in Excel and savedas a “.PRN” file to ensure the data is spacedelimited (the program will not function properlyif the data is tab delimited). The constantswill be output in the form required by the ILXLightwave temperature controller.Alternatively, the program called EasySTEIN2.EXE can be used. This program will promptfor the data to be input directly, rather thanusing a separate data file. As with STEIN2.EXE, the constants are output in a formthat is entered directly into the temperaturecontroller. The constant uncertainties are alsocalculated and displayed.The constants are output in the form used byan ILX temperature controller by scaling eachas shown below.(6) C1 C1 * 10(R)3244415534 986459363560-13(7) C2 C2 * 104(8) C3 C3 * 107The program or spreadsheet performs thisscaling so the output values can be entereddirectly into the temperature controller.Both programs use the method described byPhilip R Bevington in “Data Reduction andError Analysis for the Physical Sciences,”McGraw-Hill, New York, 1969. Matrixinversion is used to solve N simultaneousequations, where N is the number of datapairs in the data file (excluding the marker).Coefficients C1, C2, and C3 are determined byMethod 2, STEIN2.EXESTEIN2.EXE uses least-squares-fit errorreduction, so requires a greater number oftemperature/resistance readings to be taken4minimizing 2, the measure of the fit of thecurve to the data.tolerance 10k thermistor is better than 0.6 C at 50 C.Faster Method 2As discussed previously, some manufacturersprovide nominal R-T values with thethermistor. “Nominal” Steinhart-Hartconstant values can be calculated from themanufacturer’s R-T values with Method 2 ifhigh temperature control tolerances are notrequired for a particular application.Method 3, STEIN3.EXESimilar to Method 2, the third method usesleast-squares-fit error reduction to adjustthe R-T curve for a good fit. This method isintended for use when only the first twoSteinhart-Hart constants are used.The R-T data is collected and formatted thesame as described for Method 2, but theprogram titled STEIN3.EXE is used.Temperature error can be calculated usingEquation (4) with the “nominal” constantvalues and the worst-case resistance valuesfrom the tolerance rating. Results of thisexercise are shown in Figure 3 for a 1% and5% tolerance thermistor. The error is theuncertainty in the temperature based on theresistance tolerance when deriving C1, C2,and C3; it does not include any additional errorbased on the uncertainty in the resistancemeasurement.As with Method 2, a faster method usingMethod 3 can be performed by entering thenominal R-T values supplied by the thermistormanufacturer and running STEIN3.EXE. Theresulting “nominal” Steinhart-Hart constantvalues can then be entered into the LDT5525. Temperature error can be calculatedusing Equation 5 with the “nominal”constant values and the worst-case resistancevalues from the tolerance rating.As shown in Figure 3, below 50 C this error isless than about 1.5 C for the 5% tolerance10k thermistor. The error for a typical 1%As shown in Figure 4, below 50 C this erroris less than about 2 C for the 5% tolerance10k thermistor. The error for a typical1% tolerance 10k thermistor is better than 0.8 C at 50 C.Thermistor Calibration Error in Temperature with Eq (4)32Figure 4 also shows the error associatedwith the calibration for a Model 510 (10k )thermistor, used with the Model LDT-5525Temperature Controller when using Equation5. Again, this does not include any error in theresistance measurement.Oto adjust the R-T curve for a good fit. Datashould be entered into an ASCII data file inthe format shown in Figure 2.Thermistor Calibration Error ( C)6. Compile the data into a table with twocolumns: “true” temperature calculatedusing Equation 4, and resistancemeasured from the uncalibrated thermistor.15% Tol.1% Tol.0-1-2-3-20-10010203040506070Temp (O C)Figure 3. Temperature error due to thermistorcalibration error.5

Thermistor accuracy is primarily a functionof the thermistor calibration and resistancemeasurement accuracy, whereas temperaturestability depends on the controller and controlenvironment.Thermistor Calibration Error in Temperature with LDT-55254OThermistor Calibration Error ( C)32Calibrated5% Tol.1% Tol.1For more information on thermistor selection,see ILX Lightwave Application Note #2,Selecting and Using Thermistors forTemperature Control.0-1-2-20-10010203040506070Temp (O C)Copies of the programs may be obtained fromILX Lightwave free of charge through theDownloads section on www.ilxlightwave.com.Figure 4. Temperature error due to thermistorcalibration error.In ConclusionThermistor calibration, though not difficult,can be time-consuming. Therefore, it isbest to first determine the requirements ofthe application, then pick an appropriatecalibration method. The methods discusses inthis publication are summarized in Table 2.Often, the thermistor manufacturer will provideR-T values, but the accuracy of these valuesdepends on the resistance tolerance of thethermistor. When a high temperature tolerance is required, it is recommended that theseR-T values be discarded and new values bemeasured as described below.The control temperature tolerance willdecrease rapidly if the thermistor is usedoutside of the temperature range in which itwas calibrated. Temperature and resistancevalues should be made at evenly spacedincrements over a range greater than theintended range-of-operation for the thermistor.For some applications, the nominal R-T datais adequate and the Steinhart-Hart constantscan be calculated using “Faster Method 2,”described below.1. Set a nominal temperature and allow it tostabilize.2. Using a precision DMM (accuracy to aminimum of four places) read the resistanceof the reference thermistor and theuncalibrated thermistor.ILX Lightwave Model 520 uncalibratedthermistors are shipped with three-termnominal constant values as follows:C1 1.125C2 2.347C3 0.8553. Read the resistances three times beforechanging to a new nominal temperature.The three readings can be averaged if usingMethod 1, or all readings can be used ifusing Method 2 or Method 3.The two-term nominal constants, for use withthe LDT-5525, are:C1’ 0.99C2’ 2.57Table 2Summary of Calibration Methods*Also used in ILX Lightwave Model 37xx and 39xx Laser Diode Controllers.4. If using Method 1, repeat steps 1-3 for atotal of three temperature settings. Foreither Method 2 or Method 3, repeat themeasurement as many times as practical;these two methods use least-squares fitto determine the constants, and will bemore accurate with a greater number ofmeasurements.Procedure for CalculatingSteinhart-Hart ConstantsTo calculate the constants for a newthermistor, the temperature and resistance ofthat thermistor will need to be measured atseveral different temperatures covering theexpected range of operation.5. The “true” temperature in degrees Celsiuscan be determined using the constantsfor the precision thermistor and using theinverse of Equation 1, shown below asEquation 4.The following procedure requires somemethod to set and control a nominaltemperature and a calibrated precisionthermistor to reference the temperature.(4) T (C1 C2 * ln(R) C3 * ln(R)3) –1 – 273.15For all methods, it is worth noting thatthe ultimate accuracy of the constantsis dependent upon the accuracy of thetemperature and resistance measurements.6If using Method 3, the approximated Steinhart-Hart equation, disregard the term usingthe constant C3 by using Equation 5.(5) T (C1 C2 * ln(R)) –1 – 273.153

of the other two tolerances. The three factorsare related as shown in Equation 3.The Excel version of STEIN1.EXE is printed inthe Appendix.(3) Ttol Rtol / Table 1Three Methods of CalculatingConstants C1, C2, C3When a thermistor is calibrated with theSteinhart-Hart model, its temperaturetolerance over that range is improved tothe tolerance of the model. Therefore, aninexpensive thermistor calibrated to 0.02 Cwill be just as accurate as an expensive(i.e. tight tolerance) thermistor that isalso calibrated to 0.02 C over the sametemperature range.Table 1 Notes:Net AccuracyAn LDT-5948 or 5980 or other ILXTemperature Controller may be used toindependantly measure the temperaturewhen calibrating a thermistor. However,to guarantee accuracy, the instrument’sresistance measurement must be accuratelycalibrated and a previously calibratedthermistor (with the Steinhart-Hart coefficientsentered) must be used to measure thetemperature. Also, accuracy will be reducedby the temperature resolution of theinstrument, unless the temperature is queriedvia GPIB.1 Accuracy over 0 C to 50 C range; assuming temperature and resistancereadings are accurate to four places.2 Using 10k thermistor and ILX Lightwave model LDT-5910Btemperature controller.3 Using 10k thermistor and ILX Lightwave model LDT-5525 temperaturecontroller.Discussion Regarding TemperatureAccuracyThe method of thermistor calibration willdepend on the accuracy requirements forthe particular application. Table 1 shows theexpected accuracies using the three differentmethods.Thermistor RatingsManufacturers specify thermistor tolerancesin several ways, usually with the resistancetolerance (Rtol) or temperature tolerance (Ttol),and the temperature coefficient of resistance( ). The rated Rtol and Ttol are typically givenfor 25 C with additional deviation factorsfor other temperatures. The temperaturecoefficient of resistance ( ) is the percentagechange of resistance for a 1 C change intemperature, and may be specified with oneStability vs. AccuracyTemperature accuracy, which is the variancefrom true temperature, depends primarilyon the thermistor calibration. Temperaturestability, which is the invariance from the settemperature, depends on the controller designand the environment of the thermistor and TEmodule.If an LDT-5948 or 5980 is used, short-termtemperature stability of 0.001 C or better canbe achieved.2APPENDIX - Method 1 Excel SpreadsheetType in the equations as shown. Temperature readings are entered in Cells C3-C5; Resistance readings areentered in Cells F3-F5. The results are shown in Cells F9-F11, and are scaled so they may be entered into theLDT-5910B 5BCEnter Temperature Values HereDEFEnter Resistance Values HereT1 T1R1 R1T2 T2R2 R2T3 T3R3 R3T1K C3 273.15T2K C4 273.15T3K C5 273.15A1 LN (F3)A2 LN(F4)A3 LN(F5)Z C11-C12Y C11-C13X 1/C7 - 1/C8W 1/C7 - 1/C9V C11 3 - C12 3U C11 3 - C13 3C3a (C17-C15*C18/C16)/(C19-C15*C20/C16)C2a (C17-C22*C19)/C15C1a 1/C7-C22*C11 3-C23*C11Results are shown he

Steinhart-Hart equation. The second form of the equation is simpler, and is used when only the fi rst two Steinhart-Hart constants are used. The Steinhart-Hart Equation The three-term Steinhart-Hart equation (Equation 1) is the most popular model used for thermistor