handbook of modern sensors

468 16 Temperature Sensors
An obvious advantage of this model is a need to calibrate a thermistor at only one point
(S 0 at T 0 ). However, this assumes that value of β m is known beforehand, otherwise a
two-point calibration is required to find the value of β m :
β m = ln(S 1/S 0 )
(1/T 1 − 1/T 0 ) , (16.19)
where T 0 and S 0 , and T 1 and S 1 are two pairs of the corresponding temperatures and
resistances at two calibrating points on the curve described by Eq. (16.18). The value
of β m is considered temperature independent, but it may vary from part to part due to
the manufacturing tolerances, which typically are within ±1%. The temperature of a
thermistor can be computed from its measured resistance S as
( 1
T = + ln(S/S ) −1
0)
. (16.20)
T 0 β m
The error of the approximation provided by Eq. (16.20) is small near the calibrating
temperature, but it increases significantly with broadening of the operating range (Fig.
16.7).
β specifies a thermistor curvature, but it does not directly describe its sensitivity,
which is a negative temperature coefficient, α. The coefficient can be found by
differentiating Eq. (16.18):
α r = 1 dS
S dT =− β T 2 . (16.21)
It follows from Eq. (16.21) that the sensitivity depends on both β and temperature.
A thermistor is much more sensitive at lower temperatures and its sensitivity drops
quickly with a temperature increase. Equation (16.21) shows what fraction of a resistance
S changes per degree of temperature. In the NTC thermistors, the sensitivity
α varies over the temperature range from −2% (at the warmer side of the scale) to
−8%/ ◦ C (at the cooler side of the scale), which implies that an NTC thermistor is a
very sensitive device, roughly an order of magnitude more temperature sensitive than
a RTD. This is especially important for applications where a high output signal over a
relatively narrow temperature range is desirable. An example is a medical electronic
thermometer.
16.1.3.1.2 Fraden Model
In 1998, the author of this book proposed a further improvement of the simple model
[4]. It is based on the experimental fact that the characteristic temperature β is not a
constant but rather a function of temperature (Fig. 16.6). Depending on the manufacturer
and type of thermistor, the function may have either a positive slope, as shown
in Fig. 16.6, or a negative one. Ideally, β should not change with temperature, but that
is just a special case that rarely happens in reality. When it does, the simple model
provides a very accurate basis for temperature computation.
It follows from Eqs. (16.16) and (16.17) that the thermistor material characteristic
temperature β can be approximated as
β = A 1 + BT + A 2
T + A 3
T 2 , (16.22)