handbook of modern sensors

470 16 Temperature Sensors
and the manufacturing process, so it may be considered more or less constant for a
production lot of a particular type of a thermistor. Thus, it is usually sufficient to find
γ for a production lot or type of a thermistor rather than for each individual sensor.
By substituting Eq. (16.23) into Eq. (16.16), we arrive at a model of a thermistor:
ln S ∼ = A + β m [1 − γ (T b − T )]
. (16.26)
T
Solving Eq. (16.26) for resistance S, we obtain the equation representing the thermistor’s
resistance as a function of its temperature:
S = S 0 e β m[1+γ (T −T 0 )](1/T−1/T 0 ) , (16.27)
where S 0 is the resistance at calibrating temperature T 0 and β m is the characteristic
temperature defined at two calibrating temperatures T 0 and T 1 [Eq. 16.19)]. This
is similar to a simple model of Eq. (16.18) with an introduction of an additional
constant γ . Even though this model requires three points to define γ for a production
lot, each individual thermistor needs to be calibrated at two points. This makes the
Fraden model quite attractive for low-cost, high-volume applications which, at the
same time, require higher accuracy. Note that the calibrating temperatures T 0 and T 1
preferably should be selected closer to the ends of the operating range and for the
characterization, temperature T B should be selected near the middle of the operating
range. See Table 16.3 for the practical equations for using this model.
16.1.3.1.3 The Steinhart and Hart Model
Steinhart and Hart in 1968 proposed a model for the oceanographic range from −3 ◦ C
to 30 ◦ C [5] which, in fact, is useful for a much broader range. The model is based on
Eq. (16.16), from which temperature can be calculated as
[
T = α 0 + α 1 ln S + α 2 (ln S) 2 + α 3 (ln S) 3] −1
. (16.28)
Steinhart and Hart showed that the square term can be dropped without any noticeable
loss in accuracy; thus, the final equation becomes
[
T = b 0 + b 1 ln S + b 3 (ln S) 3] −1
. (16.29)
The correct use of Eq. (16.29) assures accuracy in a millidegree range from 0 ◦ Cto
70 ◦ C [6]. To find coefficients b for the equation, a system of three equations should be
solved after the thermistor is calibrated at three temperatures (Table 16.3). Because of
the very close approximation, the Steinhart and Hart model became an industry standard
for calibrating precision thermistors. Extensive investigation of its accuracy has
demonstrated that even over a broad temperature range, the approximation error does
not exceed the measurement uncertainty of a couple of millidegrees [7]. Nevertheless,
a practical implementation of the approximation for the mass produced instruments is
significantly limited by the need to calibrate each sensor at three or more temperature
points.