Linearize thermistors with new formula

Mark Bieger

EDN

I often have to get a reasonably accurate temperature measurement for a laser, transistor, or printed circuit board. I usually use a thermistor for my temperature sensor because they are inexpensive. However, the thermistor’s nonlinear resistance characteristic makes accurate temperature conversion complex. To minimize this difficulty, I often linearize the thermistor’s resistance characteristic by placing the thermistor within a resistor divider. This linearized response is simpler to convert to a temperature value than the thermistor’s raw response. To obtain an optimum level of linearity, I have derived a pair of formulas that are useful in determining component values (R_S and R_P) for this common linearization circuit.

Figure 1 shows the thermistor linearization circuit that I am addressing in this Design Idea. The temperature of the thermistor is linearly related to the output voltage (approximately).

Figure 1.	Two-resistor thermistor linearization circuit.

The design process begins by picking a temperature, which I call the inflection temperature T_I, at which we want the flattest possible transfer function (V_OUT/V_IN). My design task is to compute values for R_S and R_P given T_I and the resistor ratio

at T_I.

Three parameters, R₀, β, and T_REF, are often used with Eq. 1 to model the thermistor’s resistance versus temperature characteristic

(1)

I determine R_S and R_P by setting the second derivative of V_OUT/V_IN in Figure 1 to 0 at temperature T_I. After much algebra, I obtained the two formulas shown in Eq. 2.

(2)

Not all ratios are possible with passive components – ratios less than

result in negative R_P values.

Figure 2 shows a graph of the linearized thermistor voltage transfer function for a common thermistor.

Figure 2.	Graph of the linearized thermistor transfer function.

I have put together an Excel spreadsheet and a Mathcad worksheet that models this data, downloadable below.

ZIP archive of spreadsheet & worksheet (please use "Save as" and remove the dummy .txt extension before saving)