Page 177 - Mechanical Engineers' Handbook (Volume 2)

166 Temperature and Flow Transducers

with the required accuracy. For the sensitivity mentioned above, 50 ppm corresponds to
about 1 mK. Measurements in the 1–100-mK range are more typical of standards room
practice than of field measurements, yet some applications require this precision.

4.6 Determining Temperature from Resistance

Experimental evidence shows that the resistance of a thermistor varies inversely and nearly
exponentially with temperature. This suggests expressing the resistance–temperature rela-
tionship as a polynomial in ln(R):

1
2
3
A A ln(R) A ln(R) A ln(R) A ln(R) N (9)
T 0 1 2 3 N
Note the R is dimensional in this equation; hence the values of the constants would
depend on the units used for R.
A form retaining terms through the cubic, sufficiently accurate for small ranges in tem-
perature, was used by Steinhart and Hart. 44 Their results indicated that the second-order
terms had no effect, but later work has shown some advantage in retaining these terms.
However, the Steinhart–Hart equation is still widely used:
1
A A ln(R) A [ln(R)] 3 (10)
T 0 1 3
Arguments concerning the need for dimensional homogeneity led Bennet 45 to propose
a dimensionless form which is independent of the units used:

R
1 1 A ln A ln 2 A ln N
R
R
(11)
T T 0 1 R 0 2 R 0 N R 0
Justification for this form can be seen in the near-linearity of Fig. 24.
This general form has given rise to several simplified forms. For a first approximation,
only one term is needed:
R R exp A 1
1
1
0
T 0 T (12)
This form reveals the exponential nature of the thermistor resistance response to changes
in temperature.
46
Seren et al. compared four different forms of the calibration equation for goodness of
fit to a data set covering a range of 200 K, published by Bosson, Guttmann, and Simmons, 47
and accepted the data set as reliable. Their results are shown in Table 8. The following
calculations were used to arrive at those results:

Table 8 Accuracy of Least-Squares Fits of Four Different Polynomial Expressions for Thermistor
Calibration
Form of Equation for 1/T SRE RMS Error Mean Error
1. A 0 A 1 ln(R) 0.00601 1.999 0.1925
2. A 0 A 1 ln(R) A 2 [ln(R)] 2 0.000428 0.1353 0.0026
2
3. A 0 A 1 ln(R) A 2 [ln(R)] A 3 [ln(R)] 3 0.000139 0.0392 0.00008
4. A 0 A 1 ln(R) A 3 [ln(R)] 3 0.000138 0.0380 0.00001
Source: Reference 47.