# NTC Thermistor Linearization  Written by

Thermistors are a low-cost way to measure temperature, especially if you are not looking for crazy levels of precision. They are simply a resistor whose value varies with temperature. The effect was discovered by Michael Faraday in 1893, although commercial components were not available until a hundred years after this. Faraday observed the effect in silver sulphide, however modern thermistors tend to be made of ceramic or polymer materials.

The most common types have a negative temperature coefficient (NTC) which means their resistance decreases with increasing temperatures. The biggest disadvantage of thermistors compared say to a more expensive resistance temperature detector (RTD) is that their temperature-resistance characteristic is not linear. FIGURE 1. The resistance of a typical 10K NTC thermistor ranges from 150K at -25°C to a few hundred ohms at 125°C, although the curve is far from linear. It is in fact a negative exponential relationship. This is why we need to apply some form of linearization. (CLICK TO ENLARGE)

Figure 1 shows the relationship between temperature and resistance of a typical device. This is a Vishay NTCALUG39A103G model with a nominal resistance of 10kW. It is bult into an aluminium lug suitable for mounting to a heatsink, for example. The datasheet provides the nominal resistance which is measured at a nominal temperature (usually 25°C) along with a figure for Beta (b) of 3,984 Kelvin. Together, these figures describe the resistance-temperature relationship via the nasty exponential function below.

In this equation RT is the resistance at any given temperature T, R0 is the nominal resistance mentioned above, andT0 is the temperature that the nominal resistance is measured. Note that all temperatures are in Kelvin, so you need to add 273 to the temperatures in Celsius to use this equation.

To go the other way and calculate the temperature given the resistance, it helps to introduce another variable, R∞. This is the resistance at T = ∞, which makes no sense practically, but reduces the above equation to:

This is 15.6 × 10-3Ω for our device. We can use this to calculate the temperature given the resistance of our thermistor:

In practice, you will probably convert the thermistor resistance to a voltage using a voltage divider as shown in Figure 2. By putting the NTC thermistor in the upper part of the divider we get a voltage that increases with temperature as shown in Figure 3. We have used a 10kW series resistor and a 10.0V source. The figure shows the output voltage takes an S-shaped profile around the red dotted linear best fit line. FIGURE 2. A typical application will use the thermistor in a voltage divider. Putting the thermistor in the upper position results in a voltage that increases with temperature. (CLICK TO ENLARGE) FIGURE 3. The solid curve shows the output of the voltage divider of Figure 2 and the dotted curve is the line of best fit. The output voltage follows and S-shaped curve with significant deviation from the ideal linear voltage-temperature relationship. Note that at mid-scale (around 45°C) that the error is particularly high. (CLICK TO ENLARGE)

If we just wanted to detect an overtemperature condition on a heatsink, this would be all we need to do. We would just use a comparator to detect the voltage rising above a predetermined threshold.  If we wanted to know the actual temperature, we will need to linearise this curve. We could do this by digitising the voltage and linearising in firmware with either a look-up table or a piecewise linear approximation.

If we want to stay in the analog world, we can improve the linearity of the voltage with respect to temperature with the addition of a single resistor in parallel with the thermistor as shown in Figure 4. FIGURE 4. Adding a resistor in parallel with the thermistor improves the voltage-temperature linearity. We chose the value of Rp to be similar to the resistance of the thermistor at 45°C to maximize the linearity across the temperature range. (CLICK TO ENLARGE)

If you choose the parallel resistor Rp such that its value is about equal to the thermistor value at the centre of the range of interest, you will get a close-ish approximation of a linear voltage-temperature relationship as shown in Figure 5. In my case I chose Rp to be 4.3kW—around the value of the thermistor at 45°C. You can see the curve much better approximates the red dotted linear best fit line. FIGURE 5. The addition of a parallel resistor improves the linearity significantly. This improvement comes at the expense of the dynamic range of the signal which is now only about 3V compared to the 9V we saw in Figure 3. (CLICK TO ENLARGE)

Of course, this improvement in linearity comes at the expense of a reduced dynamic range. You can optimize the dynamic range and linearity by careful adjustment of the series and parallel resistors. You will also get better results if you are only interested in a narrow range of temperatures. Like everything in electronics, it’s all about trade-offs and compromises.

References:

“Thermistor.” In Wikipedia, June 7, 2021. https://en.wikipedia.org/w/index.php?title=Thermistor&oldid=1027407366.

“NTCALUG03A / LUG39A Mini Lug Series” Vishay, https://www.vishay.com/docs/29114/ntcalug3.pdf

“Thermistor App Notes” Siemens Matsushita Components, https://users.physics.unc.edu/~sean/Phys351/techresource/data_sheets/thermistor%20app%20notes.pdf

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##### Andrew Levido
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Andrew Levido (andrew.levido@gmail.com) earned a bachelor’s degree in Electrical Engineering in Sydney, Australia, in 1986. He worked for several years in R&D for power electronics and telecommunication companies before moving into management roles. Andrew has maintained a hands-on interest in electronics, particularly embedded systems, power electronics, and control theory in his free time. Over the years he has written a number of articles for various electronics publications and occasionally provides consulting services as time allows.  