How accurately can you in principle measure temperature with an RTD or thermistor?

If we push a current $I$ through the resistor we'll get a voltage of $U=RI$ across it. Now as the temperature changes the resistance will change by $\Delta R = \alpha R \Delta T$ where $\alpha$ is the temperature coefficient of the sensor. This will give us a signal

On the other hand the Johnson noise across the resistor will be $U_n = \sqrt{4 k_b T R B}$ where B is the bandwidth, and we get a signal-to-noise ratio of

The noise-quivalent-temperature (NET) can be defined as $\mathrm{SNR} = 1$ or

Here we can identify $P=UI=R I^2$ as the power dissipated in the resistor and simplify to

Here's a table with some common values for pt100 and 10k NTC thermistors. The sensitivity $\alpha$ is determined by the sensor type. What limits $P$ is self-heating of the sensor which probably should be kept to a few milli-Kelvins in most precision applications. Thermistors with their higher sensitivity are an obvious choice for high-resolution applications, but the lower $\alpha$ of a pt100 sensor can be compensated with a larger $P$ since most pt100 sensors are physically larger and thus have lower self-heating. pt100 sensors require 4-wire sensing, slightly more complex than a 2-wire measurement which is OK for a thermistor.

 Sensor Resistance Sensitivity (divide by R to get alpha!) Dissipated Power P Noise-Equivalent-Temperature (1Hz bandwidth) pt100 100 Ohms 0.391 Ohms/C 100 uW (I=1mA) 3 uK NTC Thermistor 10 kOhms -500 Ohms/C 9 uW (I=30uA) 0.9 uK

I conclude that it is not entirely obvious how to choose between a pt100 and a 10k thermistor. The thermistor is intrinsically more sensitive, but with good thermal contact to its surroundings self-heating in a pt100 sensor can be minimized and the same noise-requivalent-temperature achieved. In any case it looks like Johnson noise limits resolution to 1 uK or so in a 1 Hz bandwidth. If we AD-convert the voltage at 24-bit resolution (16M states) we can get a reasonable measurement range of ~32 K by matching 1 LSB = 2 uK.

Does anyone know of similar back-of-the-envelope calculations for other sensors (Thermocouples, AD590)?

## 1 Comment

1. I think that the shot noise of the polarizing current Ipol (current noise) 2*Q*Ipol should be also mentioned. This current noise is larger than the Johnson-Nyquist noise (voltage noise) since R*Ipol = 100mV (52mV rule) However, if the polarization current Ipol is provided through the amplifier, it can be decorrelated (see bolometers set-ups)
In practicle, the voltage noise of the amplifier will be dominant. This voltage noise has a certain1/f component. In one word, it seems difficult to have a noise less than 10µ°K/Root(Hz)

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