Noise equivalent temperature in an RTD/Thermistor

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

\Delta U = I \Delta R = I \alpha R \Delta T

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

 \mathrm{SNR} = { I \alpha R \Delta T \over \sqrt{4 k_b T R B} }

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

 \Delta T_{NET}= \sqrt{ {4 k_b T R B \over R^2 I^2 \alpha^2 } }

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

 \Delta T_{NET}= \sqrt{ {4 k_b T B \over P \alpha^2 } }

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)?

2 thoughts on “Noise equivalent temperature in an RTD/Thermistor”

  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)

  2. In addition NTC thermistors are typically fabricated from metal oxides which have their own set of noise characteristics. Lastly, even if there is no metal oxide material noise if there is thermal noise in the system then that will be exhibited as voltage noise as transduced by the sensor itself.

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