# Linearize thermistors with new formula

-March 03, 2014

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 (RS and RP) 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 TI, at which we want the flattest possible transfer function (VOUT/VIN). My design task is to compute values for RS and RP given TI and the resistor ratio µ=VOUT/VIN at TI.

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

(Eq. 1)

I determine RS and RP by setting the second derivative of VOUT/VIN in Figure 1 to 0 at temperature TI. After much algebra, I obtained the two formulas shown in Eq. 2.

(Eq. 2)

Not all ratios are possible with passive components – ratios less than    result in negative RP 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

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