Stuart
R.
Linearized Thermistor Thermometer
Gunn
Lawrence L~vermoreLaboratory University of California Livermore, 94550
Bridges for Calorimetry
Thermistors are extensively used for precise measurement of small temperature changes in calorimetry and other fields, particularly for reaction calorimetry in the vicinity of 25'C. Usually the thermistor is incorporated in a manually operated Wheatstone bridge, or some variation thereof, with a null detector; the time-temperature curve is recorded by noting the hridge setting and small residual deflections of the null detector a t fixed time intervals or by noting the times a t which the null is attained for fixed hridge settings. However, the present availability of electronic amplifiers having gain of high linearity and stahility and a high input resistance offers the possibility of using an unbalanced hridge without changing the value of any arm during the course of an experiment. The bridge output is amplified, fed to a digital voltmeter, and recorded by a printer, card punch, tape punch, or other data logging system. No operator attention is required, and a larger number of time-temperature points may he recorded with consequent improvement of statistical precision. In calculating the corrected temperature rise in isoperibol calorimetrv it is referable to e m.~ l o va scale which is a linear function of temperature. The resistance of a thermistor is however an ex~onentialfunction of absolute temperature R = R, expB(T-' - To-') (1)
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Nevertheless. under certain conditions. a "corrected resistance rise" calculated using a thermistor resistance versus time curve is adeauatelv to heine- . oro~ortional to the " close . . corrected temperature rise.1 The output of an unbalanced bridge is also in general a quite nonlinear function of temperature. Of course, if the corrected temperature rise is computed by machine methods, it is easy to incorporate in the program a conversion of each voltage or resistance point t o temperature; even approximate values of the thermistor and hridge parameters may give adequate linearity of the resulting temperature scale. But it is often convenient to use a measuring system whose output is nearly a linear function of temperature. It seems to he not generally realized that this can he achieved, over limited ranges of .temperature, by appropriate selection of the hridge parameters; the purpose of this article is to describe certain confirmrations to achieve this end. The circuit is illustrated in Fieure 1. For all of the Dresent examples, R1 is a thermistor having a resistance of 2000 ohms a t 25°C and B is 3500°K; the resistance is assumed to follow eqn. (1) exactly. V is a constant-voltage source such as to give a current of 100 pA, or a power dissipation of 20 pW, in the thermistor at 25°C. A variety of sets of circuit parameters are listed in the table. Certain values of Rz and R3 were selected for illustrative purposes, and Rn necessarily equals 2000 RaIRz to balance the bridge at 25°C. Sz5 is the sensitivity d E / d T a t 25°C in microvolts Der demee: .. . it is assumed that the system measurThis work was performed under the auspices of the U.S. Atomic Energy Commission. lGunn, S. R., J. Chern. Thermodynamics,3,19(1971).
Circuit Parameters
Set
R2
RS
RI
Rr
Rg
V
Sza
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Figure 1. The bridge circuit.
ing E has infinite input impedance. Comparing sets 1, 3, 6 and 7 it is seen that Szs increases as Rz increases; however, the input resistance seen by the amplifier also increases. Comparing sets 2, 3, 4, and 5 it is seen that Szs is independent of the value of R3; as Ra decreases, the input resistance seen by the amplifier decreases but the drain on the power supply and the heat dissipated in Ra and Rq increase. The variation of ST with temperature is shown in Figure 2. At a given value of Rz, the shape of the curve is independent of Ra. Sets 8-17 are optimized to give S2n.5 = S Z ~ . JS ; Z ~ . ~ / S ~ ~ is 0.999903. Sets 14-17 are designed to make use of a mercury cell as a convenient constant-voltage source. With Rs = m , the circuit cannot be optimized with Rz > 1417.75, and for the configuration of sets 14-17 it cannot be optimized with R2 over about 1250 without using excessively ~ T for all low values of R5 Re. The plots of S T / S Zversus of these sets are identical (Fig. 2). For other values of B, 1 - S T / S is~ proportional ~ to B 2 . If a constant-current rather than constant-voltage sup& ~ Tall ply is used with sets 1-7, the plots of s ~ / versus have negative slopes throughout the 2030°C region, and the slopes are more negative than those using the constant-voltage supply. If, a second 2000-ohm thermistor is substituted for RJ in set 3, the result is a Maier transposed-arm bridge. For
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Volume 50, Number 7, July 1973
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Figure 2 . Bridge sensitivity versus temperature for the indicated sets.
this 825 is twice that of set 2 and the plot of S T / S versus ~~ T is the same. This system cannot be linearized by configurations analogous to sets 8-13 or 14-17. Similar optimizations may be performed for other thermistor parameters, temperature ranges, and power dissipations; the necessary calculations are easily pro-
grammed for machine calculation. In particular, i t may he desirable to use higher thermistor power dissipations in unbalanced bridges with automatic recording than in ordinary manually operated bridges. The average temperature of a thermistor is higher than that of the hody being measured by an amount which is directly proportional to the power dissipated in the thermistor and to the thermal resistance between the thermistor and the hody. In the measurement of temperature changes this elevation does not matter insofar as it is constant. In the automatic recording bridge, the current is not being intermittently turned off or reversed to check for thermal voltages, so there is no intermittent cooling and reheating of the thermistor; and a higher current reduces the relative effect of unknown thermal voltages in the system. For all sets 8-17, the power dissipation in the thermistor increases from 19.14 fiW at 20'C to 20.48 FW at 30PC; self-heating was not considered in the calculations. For calorimetry, and for some other purposes, it is only the nonlinearity of the temperature elevation which matters. The actual temperature elevation of a thermistor with a given measuring current may he estimated by placing the thermistor in a calorimeter or other hody at constant or slowly varying temperature and measuring the resistance at different currents (or the bridge unbalance a t different supply voltages, if the resistances of other components of the bridge are not current-sensitive). The resulting plots of the resistance or output versus P or Cn will be nearly linear and can be used to deduce a thermal resistance deg/W.
