Deep-Sea Research, 1968, VoL 15, pp. 497 to 503. Pergamon Press. Printed in Great Britain.

INSTRUMENTS AND METHODS

Calibration curves for thermistors

JOHN S. STEINHART* a n d STANLEY R. HART*

(Received 15 April 1968)

Abstraet--A function for use in interpolating thermistor calibrations has been found which is suitable
for use with a wide variety of thermistors for ranges of a few degrees to a few hundred degrees.
It is
T -~ = A + B log R -F C (log R) 3
where T is the Kelvin temperature, R the resistance, and A, B and C constants. This function
possesses a number of desirable properties and fits thermistor data considerably better than many
such functions now in use.
INTRODUCTION
TrmRMISTORS,in recent years, have become by far the most common devices for the precision measure-
ment of temperature in limnology and oceanography and their use is widespread in many other
fields as well. Because they are inexpensive, while still retaining high resolution of temperature,
they are commonly used in the measurement of heat flow, small temperature differences in currents,
and in the study of long-term temperature variation in deep water as well as the microstructure
of the temperature gradient. These applications and others frequently require precision of 0.001°C
and multiple sensors which are to be intercompared. Calibration procedures to maintain laboratory
precision sufficient to produce field measurements to 0.001°C are well known and demanding, but
the fitting of curves to calibration data remains a difficult problem. It is to this problem we address
ourselves.
The customary representation for temperature dependence of resistivity, p, in a semiconductor is
p-1 _-- F (T)exp (-- AE/2kT) (1)
where AE is the energy gap between the valence band energies and the conduction band energies,
T is the Kelvin temperature, and k is Boltzman's constant. F (T) is itself a function of temperature;
thus even in simple compounds the theory does not provide a suitable, explicit law to which calibration
data may be fitted. ROeERTSONet aL (1966) point out that the spinel structure of thermistors usedin
precision temperature measurement does not have simple, sharply defined energy bands and the AE
term is consequently in doubt, even if the composition is accurately known.
Several attempts have been made to find empirical " l a w s " relating temperature to resistance
for thermistors (BEcKFm, GREEN and PEAZSON, 1946; BOSSON, GUTMA~r~ and SIMMONS, 1950;
ROS~TSON et al., 1966). We find it more fruitful to treat the problem simply as one of curve fitting
and searched for a relationship which would fit the data as well as possible but was still simple
to use. To this end we set up criteria for choice of a relationship which would be most desirable for
interpolation of temperature measurements.
These criteria, with reasons for their choice, are as follows :
(I) A single smoothly varying relationship for the entire temperature range. A common practice
at many institutions consists of fitting and smoothing a series of functions (or the same function
several times) for the total range of temperature involved. This technique can produce very close
fits but may disguise systematic deviations in some subregion of the temperature span. Ideally
these deviations should be revealed as consistent residuals.
*Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washintqon,
D.C., U.S.A.
497
498 Instruments and Methods

(2) Absence of the "plus-minus-plus" effect. This is another way of saying that a proposed
fitting function must have the right shape. For example, in a very small temperature span the simple
exponential relationship log R = A + BT -1 (A, B are constants) will produce satisfactorily small
residuals but the residuals will have a consistent bias of sign (plus-minus-plus) and for experimental
data this may be the only clue that the curve is simply the wrong shape for the data even though
the residuals are small.
(3) The Kelvin temperature an explicit function of the resistance. This criteria is desirable because
the fitted function will be used to calculate the temperature for a measured resistance. In a computer
this is not a serious problem, although it would be a slight advantage in ease of calculation and in
time used, but should it be necessary to calculate a few temperatures on a desk calculator or slide rule,
algebraic reduction of implicit formulae can be very time consuming.
(4) A stable, relatively simple mathematical form. Simplicity of form is obviously desirable
on aesthetic grounds, but is also required so that it may be easily determined that the fitted calibration
curve does not contain unwanted maxima, minima, points of inflection or any singularities in the
region of interest. Further, a chosen function ought not to change much if the number of points used
in fitting is changed. Of the many kinds of functions used in curve fitting, some have the unfor-
tunate property of great sensitivity to addition or removal of a point in a particular region. It is
this property we wish to avoid.
(5) The chosen function must admit of linear fitting procedures. This property is desirable so
that least squares procedures may be used and goodness of fit compared, even if the statistics of fit
are not formally valid as a maximum likelihood estimate.
In no sense are we proposing in this way to find a " l a w " of thermistor behavior. The search
is for a eatisfactory linearization of the problem satisfying as many of the above criteria as possible.
If such a function can be found it can be tested for suitability as an empirical law by examining how
well it is able to extrapolate values outside the calibration range. Failure to predict accurately
in extrapolation would show that a function was a poor " l a w " no matter how well it worked for
intm~lation.

DATA

As data for study of the fitting functions, we used calibration data on about twenty thermistors
over a 7 deg C range (0°--7°C), two thermistors over a 35 deg C range (0°-35°C), and pubfished
data of BossoN, G ~ and S ~ o N s (1950) for two thermistors over ranges of 170 deg C and
200 deg C (hereafter referred to as BGS data). These results were rechecked for our own thermistors
with two sets of later calibration data, and tried on some calibration data kindly supplied to us
by other workers (SIMMONS, private communication : YessoP, private communication). All results
were in general asreement : rims for purposes of illustration only three sets of representative data
will be discussed (Table I).
Table 1. Pertinent facts for the thermistors.

T3 $4 BGS

Maker Fenwall Fenwall Western Electric

Identification G244H K824A 4756-12 No. 1 material
Type Glass bead Glass bead Unknown
Calibration range 0°-7°C 0°-35°C - 68°-134°C

Our calibration data were obtained from a laminar flow constant temperature bath which will
maintain -4-0.002°C for 10-20rain and 4-0.0004°C for 2 rain or more. Calibration was against a
quartz oscillator thermometer with a resolution of 0,0001 deg C, and an estimated precision of
-4- 0.0003 deg C. For T3 the quartz oscillator and the thermistor were inserted in the same copper
block within the constant temperature bath. The time constants of the thermistor assembly and the
quartz thermometer are both about 1 sec. For thermistor $4 two different quartz oscillators
were employed during the calibration and the copper block was not used. The estimated precision
for this calibration set is -4- 0.002 deg C.
 Instruments and Methods 499

Absolute accuracy is much poorer than these figures but is not important for these tests. Non-
linearity of frequency vs. temperature could be sim~ificant and at these levels o f precision is dimcult
to assess. However, we used three different oscillators during the tests a n d found n o significant
difference in the residuals which could be interpreted as deviations from lincarity.

PROCEDURE
With equation (l) as a guide we examined about lO0 different relationships between resistance
and temperature with from two to five fitted constants. In each case a multiple regression program
was used to test the relationship against several sets o f calibration data. We soon found that criteria
(3) could be satisfied a n d attention was concentrated o n functions which were explicit functions o f
temperature. O f four or five reasonably good fits, the one consistently the best is
T -1 = A + B log R + C (log R) a (2)
where T - t is the inverse Kelvin temperature, R is the resistance, and A, B, and C constants to be
fitted. F o r most experimental data the resistance (or the temperature) may not enter a n expression
more than twice without encountering serious computational difficulties. This results from the
determinant of the coefficient matrix in the regression analysis becoming numerically a very small
number, a n d shows that, even though the terms (of a power series for example) may be formally in-
dependent, the presence o f real scatter in the data makes the terms appear dependent for fitting
purposes. F o r data of very high quality and wide temperature span three or occasionally four entries
o f resistance may be used a n d we determined from these cases that the addition o f the (log R)s
term in equation (2) degraded the fit.
RESULTS
The fit o f equation (2) to thermistors T3 a n d $4, listed in Table 1, is shown in Table 2. The
numerical coeffacients are given in the Appendix. Clearly the fit is good and the mean residuals
are approximately the same size as the experimental precision.

Table 2. Results of fitting equation (2).

Thermistor T3 Thermistor $4
Obs-est Obs-est
Robs Yobs Te,t (°C Robs Tobs Test (°C
(VA -x) (°C) (°C) residual) (VA -x) (°C) (°C) residual)
6304.8 -- 0.0046 -- 0.0005 -- 0-00002 5088-45 -- 0"0070 -- 0"0037 -- 0"00327
5716.8 2.1856 2.1857 0-00006 5087"9 0.0000 -- 0.0011 0.00117
5394.0 3.5004 3.4999 0.00047 4581"9 2"4845 2"4812 0"00329
5325.5 3.7905 3.7904 0.00017 4397"4 3"4645 3"4657 -- 0"00121
5277.4 3.9970 3.7904 0-00014 4357"1 3.6864 3.6870 -- 0"00067
5215.1 4-2675 4.2675 0.00000 4247.5 4"3032 4"3011 0"00202
5168.6 4.4715 4.4719 -- 0.00038 3813"1 6-9282 6"9278 0.00035
5046.7 5"0174 5"0179 -- 0-00045 3504"3 9-0130 9"0138 -- 0.00080
4663"8 6"8358 6"8357 0-00005 3236.5 11"0004 11"0019 -- 0.00151
4616.0 7.0747 7"0746 0.00009 2880.7 13"9579 13"9582 -- 0-00032
2569"1 16"9169 16"9163 -- 0"00053
2293"3 19.9040 19.9034 0"00058
1999"7 23"5792 23.5785 0.00066
1787"6 26.6461 26-6466 -- 0.00054
1573-7 30"2011 30"2019 -- 0"00083
1454"5 32"4361 32"4364 -- 0-00033
1334"6 34"9111 34"9102 0.00087
Mean residual Mean residual
0.00018 0.00111
without regard without regard
to sign to sign

Table 3 shows the results of fitting equation (2) to the Bosson, G u t m a n n a n d Simmons data.
To show the degree of improvement, the results from equation (2) are compared with the " l a w "
proposed by Bosson, G u t m a n n and Simmons a n d presently in use by ROS~TSON et al. (1966).
 O

Table 3. Comparison of fits for two equations.

B G S equation Equation (2)

logR=A + B/(T + O) T -l=A+BlogR+C(logR) a
100 (Rest -- Robs) 100 Tab s -- Test)
Observed temperature Observed resistance % discrepancy = Robs % discrepancy --
Tobs
(°K) ( V A -~) (%) (%)

204"71 10,558,700 + 0"61 -- 0"011

209"17 7,368,300 + 1"06 + 0"036
216"95 4,130,t00 -- 0"18 -- 0"015
224"16 2,465,400 -- 0"36 -- 0.011
232"54 1,398,800 -- 0-51 -- 0.010
240-05 864,970 -- 0.56 -- 0.009
247-74 540,510 -- 0.28 + 0.010
257-18 315,040 -- 0-44 -- 0-005
266"48 190,430 -- 0"27 + 0.001
276-34 115,210 -- 0"05 + 0-007
286.27 71,585 + 0-12 + 0.011 Ca.
296-35 45,383 + 0"37 + 0.020
308.54 27,163 + 0"31 + 0-006
320.24 17,104 + 0-44 + 0"010 O
331'57 11,289 + 0.04 -- 0.027
344.07 7297.4 + 0-19 -- 0"014
357"16 4761"I + 0"25 -- 0.004
369"36 3282-5 + 0"10 -- 0.008
382"29 2263"8 0.00 -- 0.002
394-74 1615.7 -- 0.09 + 0-010
407-17 1177"6 -- 0"37 + 0.007

Standard relative error, BGS e q u a t i o n : [Z' (Rest ~llRObs)2/R~] ½ 0"00395

Standard relative error, e q u a t i o n ( 2 ) : [Z'(Test --Tobs)e/Tobs~]½ = 0 " 0 0 0 1 6

 Instruments and Methods 501

Because of the different form of the two equations quantitative comparisons are not exact but the
improvement in fit of equation (2) over the BGS equation is a factor of 10 or more. The mean residual
for the equation (2) fit is 0.03°K and is about the same as the experimental precision quoted by Bosson
et al. for their data ; (temperatures 4- 0.02 deg K ; resistance -)- 0.02 ~o). From all three thermistors
we thus conclude that it is not possible to find an equation that fits more closely unless more precise
data are available. We did find in our search several relations that fitted almost as well, but none
was consistently good that also satisfied criterion 3.
Thus far we have clearly satisfied criteria 1, 3 and 5. Inspection of Tables 2 and 3 indicates that
there is no marked plus-minus-plus effect in the residuals. There is a sign consistency for the BGS
residuals, which can be completely changed by refitting after removing the points at 209°K and
331°K. In any case the effect is extremely small.

INTERPOLATION AND EXTRAPOLATION

To satisfy the remaining one of our criteria we examined the behavior of the fitted curves as
successive deletions of experimental points were carried out in an arbitrary manner, but preserving,
as far as possible, the total range of calibration. The results of this reduction in numbers of observed
points are shown in Fig. 1. The clear result is that equation (2) fits the observed points well within
the limits of experimental precision even though very few points are used to make the fit. The single
set of three points for the BGS data that exceeds the experimental precision includes the poor point
at 209.17°K, which in many fitted equations appears to be in error by about twice the experimental
precision. Further examination of Fig. 1 suggests that the best experimental strategy is to extend
the calibration range outside the range in which measurements are to be made---and that only a
few points are needed to validate the fitting procedure.
A. T3
/

.0004 1 1 / I I I I |

I I I I 1 I I I
0 1 -2 3 4 5 6 7
Degrees Celsius

B, $4 1
.003 -- ,~ ~3,~

I I
10 2O 3O
05 - .C. Degrees Celsius _

:ii I I I I
250 300 350 400
Degrees Kelvin
Fig. l. Interpolation behavior of equation (2). The ordinate is the difference in temperature
between predicted temperatures using a small number of points in fitting equation (2) and the
predicted values using all experimental points. The numbers show the number of points used
in each fit and the hatched areas show the limits of experimental precision. Differences are with-
out regard to sign.
502 Instruments and Methods

For extrapolation the situation is quite different. Figure 2 shows the departures to be expected
when a fitted curve is used to predict temperatures outside the range over which the calibration
was done. These curves were obtained by taking small segments of the calibration data and fitting
equation (2). The fitted equation was used to predict values outside the range from which it was
derived and the predictions compared to observations. The results are plotted in terms of multiples
of the experimental precision so that the three sample thermistors may be compared. For $4 and
BGS two different sub-ranges were treated in this way and both results are shown. In general the
extrapolations are reasonable for 15 to 30 per cent beyond the calibration range. Beyond that point
they deteriorate rapidly. As suggested earlier, this also shows that equation (2) is not very useful
as a " l a w , " even though it is an excellent fitting function.

100 1----I T T

"~ 50
>

~ ~ 30

~°! .21)

~,.

~'~ lo I
I
"K I
I
,-'~ 5 ,/

T3 data
.~_ $4 data
~ 2
BGS data

0.1 0.2 0.3 0.5 1.0 2 3 5 10

Extrapolation beyond the calibration range expressedas a multiple

of the calibration range
Fig. 2. Extyapolation behavior of equation (2). Normalized curves for all three thermistors
for extrapolation beyond the range of calibration. Note that the curve flrgt appears on this
graph when it exceeds the experimental precision. See text for d=i--vhtion.

There is some suggestion in the results that the temperature--resistance relationship for thermistors
is best represented as a power series of odd powers of log R. Only the BGS data covered a sul~ient
range to test this hypothesis further. Table 4 shows the standard relative error for several multiple
term fits. The results show that the even power terms do not improve the fit so well as the odd powers.
It supports the suggestion that an exact relationship would be representable as a power series of odd
powers.
It was pointed out to us by Dr. Richard P. yon Herzen that odd power series expansions of several
trigonometric functions of log R would have the general form of equation (2). We have tested sin,
 Instruments and Methods 503

Table 4. Comparative fits of several equations involving powers of log R to Bosson, Gutmann and
Simmons data.

Equation Standard relative error

T -I = A + B log R 0.006012
T -1 = A + B log R + C (log R) 2 0.000469
T -1 = A + BlogR + C(log R) a 0.000162
T -1 = A + B log R + C (1o8 R) 2 + D (log R) s 0-000472
T -1 A + BlogR + C(log R) a + D (log R) 5 0.000091

tan, sin -x, tan -x and at least these simple examples disagree with the fitted coefficients in sign, or
by several orders of magnitude or both.
CONCLUSION
An extensive examination of calibration functions has yielded a function suitable for calibration
curves for precision thermistor temperature measurements. This function is recommended to
workers malfin8 such precision measurements as its properties have been examined for a variety
of data and a variety of thermistors.
Acknowledgements--We appreciate the opportunity to try this function on unpublished calibration
data loaned to us by Dr. G. S ~ O N S of Massachusetts Institute of Technology and Dr. A. JvAsov
of the Dominion Observatory of Canada. Copies of computer programs used in this paper may
be obtained from the authors.
REFERENCES
B.~.Cr,Jm J. A., C. B. GREEN and G. L. PearSON (1946) Properties and uses of thermistors---thermally
sensitive resistors. Trans. Am. Inst. Elec. Engrs, 65, 711-725.
BOSSONG., F. GtrrMA~ and L. M. SIMMONS,(1950) A relationship between resistance and temperature
of thermistors. J. appl. Phys., 21, 1267-1268.
ROUtTSON E. C., R. RAsvFr, J. H. SW~TZ and M. E. LtLLAma (1966) Properties of thermistors
used in geothermal investigations. U.S. Geol. Surv. Bull., 1203-B, 34 pp.

APPENDIX
The table below gives the coeffmients for equation (2) obtained by least squares procedures from
a multiple regression program used by the authors.

A B C
all × 10-8 all × 10-8 all x 10 - °

BGS 0.792008 0-231076 84'261

Western Electric
T3 1.258294 0.263120 150.370
$4 1.168483 0.280480 158.816