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)?

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)

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

© 2018

Theme by Anders NorénUp ↑