Linearization of thermistors using a new formula

There are many occasions when it is necessary to measure the temperature of a laser, transistor, or printed circuit board with considerable accuracy. Thermistors are often chosen as temperature sensors because they are inexpensive. However, the nonlinear resistance characteristics of thermistors complicate accurate temperature conversion.
To minimize this difficulty, thermistors are often placed in a resistor divider to linearize their resistance characteristics. The linearized response is easier to convert to temperature than the thermistor's original response. To achieve the best level of linearization, two formulas are derived that help determine component values ​​(RS and RP) for this commonly used linearization circuit.
Figure 1 shows a thermistor linearization circuit. The temperature of the thermistor is (approximately) linear with the output voltage.


Figure 1: Dual-resistor thermistor linearization circuit.

First, a value for the inflection temperature TI is chosen, at which we want to obtain the smoothest possible transfer function (VOUT/VIN). The design task is to calculate the values ​​of RS and RP given TI and its resistance ratio μ = VOUT/VIN.
The three parameters R0, β, and Tref are commonly used in equation (1) to determine the resistance value and temperature characteristics of the thermistor.



The values ​​of RS and RP are determined by setting the second derivative of VOUT/VIN at TI in Figure 1 to zero. After much algebra, the two formulas in equation (2) are derived.




Not all ratios are possible for passive components—ratios below 1/2

will result in negative RP values.

Figure 2 shows a linearized thermistor voltage transfer function plot for a common thermistor.


Figure 2: Linearized thermistor transfer function plot.