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