wern@utzoo.uucp (Wern Thiel) (05/27/90)
here is an exerpt from an article we published in ELECTRONICS TODAY International Nov.1980 p.71 under the title USING THERMISTORS : W. THIEL J. MACHIN AND J. KORNATOWSKI Department of Zoology University of Toronto Toronto Ontario M5S 1A1 Canada. Abstract A general equation describing the resistance- temperature characteristics of thermistors is presented together with a functional interpretation of two of its constants. The use of the equation greatly simplifies calibration procedures and promotes increased confidence in inexpensive "wide tolerance" thermistors. Introduction Thermistors are temperature sensitive resistors usually with a negative temperature coefficient. One of the advantages of using thermistors for tem- perature measurement is their small size. Another is that small temperature changes cause considerable changes in their electrical resistance. Apart from the direct measurement of temperature they may be incorporated into electronic circuits as temperature compensating devices as well as forming the sensitive elements in hygrometers, vacuum gauges, conductivity cells and anemometers. (Philips Electron Devices 1963; Gulton Industries 1965; Mortimer and Moore 1970; Fenwal Electronics 1974). Manufacturers typically supply precalibrated thermistors as well as less expensive types which have much wider tolerances (to within 20% of nominal calibration). The idea has grown up, that inexpen- sive thermistors are unreliable and require frequent and laborious calibra- tion. Our experience with the calibration and use of inexpensive thermistors in biological research has removed much of this doubt and has lead to greater confidence in a simplified calibration procedure. This calibration technique together with some practical consequences of a mathematical formulation for the temperature-resistance relationship are described below. It is hoped that this information will encourage the design and use of inexpensive thermistor instrumentation by the individual experimenter. Those who employ multiple, readily portable remote reading temperature measuring devices with a sensi- tivity in the order of of 0.5 C would benefit most from the type of instrument described. The general relationship between temperature and resistance for a thermistor is given by the following exponential equation (Philips Electron Devices 1963) R = Ae(B)T where R is resistance in ohms at temperature T. T is the temperature in C and e the base of natural logarithms. Since R equals A at zero degrees Celsius, A is the intercept and B is the slope in a semi-log plot given by B = ln(R) -ln(A) / T We had thought that inexpensive wide tolerance thermistors would follow this exponential relationship less closely. This was found not to be true. When comparing a large number of conventionally obtained calibration curves from Philips thermistors (type 205-CE/P2K2), we found that values for B were remarkably consistent. It was only coefficient A that varied considerably between individual thermistors. It appears that B is related to the composi- tion of the temperature sensitive material used in the thermistor, differing only slightly within each type and even less within one batch. Since B represents the slope it is also an indication of the sensitivity of the thermistor. The value of A is a characteristic of each individual thermistor and is therefore related to the nominal values stated by the manufacturer. The spread in A for inexpensive thermistors may be as high as 20% but more recent production technique or quality controls have resulted in generally closer tolerances. It is now frequently possible to find matching pairs in a sample of ten. Thermistors of the same material in many different physical configurations are available in a wide range of nominal values from a few ohms to hundreds of thousands of ohms at room temperature. Calibration Thermistors may be calibrated by mounting them in contact with the bulb of an accurate thermometer. Calibration runs are most conveniently performed by first packing the thermistors in ice chips and distilled water to obtain zero C. Higher temperatures are then obtained by stirring in various amounts of warm water. For the present purpose thermistor resistances were con- veniently measured with a digital ohmmeter to the nearest ohm and temperatures were measured to 0.1 C. We have found that satisfactory calibration curves for thermistors can be obtained by simply determining thermistor resistance at zero C and at one other temperature preferably at the other end of the temperature range of interest and by using Eq. (1) and Eq. (2). In this manner it is possible to derive calibration curves for any unknown thermistor. Limited extrapolation to temperatures outside the calibrated range can be made but possibly with increased error. Comparisons of single resistance measure- ments can be confidently used to check for damage. We have found that undam- aged thermistors used at biological temperatures do not change their calibra- tion with time. Since coefficient B varies slightly over wide temperature ranges, more accurate calibrations may be performed at several temperatures. In this case a somewhat better overall fit to the calibration data can be obtained with a computer least squares curve fitting program which computes the best fit for an exponential equation to all the observed values. Table 1 compares observed resistance with those predicted from a computer determined curve equation based on calibration measurements made at 10 C intervals and with those predicted from a regression line between points for zero C and 50 C. In the latter the log-linear calculations were performed on an elec- tronic pocket calculator. It can be seen that the errors in calculated resis- tance do not exceed 0.9% in either case. This is equivalent to a maximum error of about 0.4 C. TABLE 1 Discrepancies between observed and calculated resistances predicted from Eq. (1) comparing the results from a computer program using all temperature values and those from an electronic pocket calculator using only two points (zero and 50C). TEMPERATURE ACTUAL COMPUTER ERROR POCKET ERROR degree RESISTANCE BEST FIT % CALCULATOR % 0 4036 4013.93 +0.6 4036 +0.0 10 2689 2689.20 +0.0 2704 +0.5 20 1792 1800.82 -0.4 1811 +1.0 30 1201 1207.12 -0.5 1214 +1.0 40 808 808.35 +0.0 813 +0.6 50 545 541.85 +0.6 545 +0.0 Appendix I Calculation To calculate the parameters of a thermistor that was measured at two points - zero degree and 50 degrees. The resistance at zero was 4036 Ohms and the resistance at 50 degrees was 545 Ohms. First to calculate B enter R 545 press ln x 6.30079 press minus enter A 4036 press ln x 8.303 press equal -2.00222 press divide enter T 50 press equal -.0400445 answer Calculate resistance for 10 degrees enter B -.0400445 press multiply enter T 10 press equal -.4004447 press e (inverse ln x) .670022 press multiply enter A 4036 press equal 2704 answer