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Thermistor Equation and Temperature Sensing

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thedatastream

Mon Jul 20 2009, 08:32PM

Registered Member #505 Joined: Sun Nov 19 2006, 06:42PM
Location: Yorkshire!
Posts: 329

Hi folks,

Maths has never been my strong point, and I'm not sure how to interpret this equation, specifically the ln^2 and ln^3

Does ln^2 (x) mean:

ln ( ln (x) )
ln (x^2)
( ln (x) ) ^2

Or something else?

Thanks in advance
James

edit: changed thread topic to better suit contents

magnetomotive

Mon Jul 20 2009, 08:43PM

Registered Member #267 Joined: Mon Feb 27 2006, 09:44PM
Location:
Posts: 46

The squared ln should be interpreted as: ( ln (x) ) ^2

GeordieBoy

Mon Jul 20 2009, 08:44PM

Registered Member #1232 Joined: Wed Jan 16 2008, 10:53PM
Location: Doon tha Toon!
Posts: 881

Hi James,

I think lnÂ²(R/Rref) means take the natural logarithm of (R/Rref) and then square the result, but I could be wrong.

There is some info about thermistor equations on wikipedia and also here:

I have only ever used the simpler equation with the exponential and beta in it before myself. (The one shown underneath the graph on that web page.)

-Richie,

EDIT: Actually, thinking about this some more, it looks like the equation i've used is a simple linear approximation to the true thermistor response. The equation you have probably attempts to fit the true curve of the thermistor more accurately by modelling it as a polynomial in increasing powers of ln(R/Ref) The A1 B1 C1 D1 are the coefficients just like how you might have coefficients of the terms xÂ¹, xÂ², xÂ³ etc

Sulaiman

Mon Jul 20 2009, 09:58PM

Registered Member #162 Joined: Mon Feb 13 2006, 10:25AM
Location: United Kingdom
Posts: 3140

Just in case you're cosidering using a thermistor for temperature measurement;
Industrially I see a lot of thermistors used as limit detectors
but for temperature measurement, thermocouple or PT100 resistor is standard
because you will have difficulty re-calibrating when the sensor (thermistor) needs replacing, ....

Dennis Rogers

Mon Jul 20 2009, 10:20PM

Registered Member #1837 Joined: Tue Dec 02 2008, 02:20PM
Location: NYC
Posts: 65

ln is the logarithm as richie stated. It would be the proper equation but in the real world you don't need it to be that precise. This equation goes a few orders of magnitude deeper than any thermistor would be rated anyway.

thedatastream

Tue Jul 21 2009, 08:07AM

Registered Member #505 Joined: Sun Nov 19 2006, 06:42PM
Location: Yorkshire!
Posts: 329

Thanks for all the responses chaps.

I tried some of the suggestions (and all three of mine) but none of them seemed to work so in the end I cheated. I used the below equation to create a chart in Excel of resistance vs temperature.

I then added a trendline to the chart (4th order polynomial fitted the curve best) and then extracted the trendline curve from the chart.

This then allowed me to work backwards from my ADC count, through the thermistor potential divider chain to achieve a correlation between ADC count and associated temperature. This then goes into a lookup table in my micro (Arduino) to give me a quick lookup without having to do natural logs in an 8-bit micro (I don't have all day you know!)

I'm using a 2% thermistor

to sense temperature over the range 10ÂºC to 50ÂºC to regulate the temperature of a heater immersed in liquid. I'd be happy with temperature control of Â±5%, probably achieavable given the innacuracies in the system.

For interest, I tried the equation from the page Richie highlighted

And compared it with the trendline generated polynomial line. See chart

1248163497 505 FT73212 Thermistor Chart Eqn Check

It's a pretty good match between the two. I'll probably stick to the beta equation in future!

Thanks again
James

Steve Conner

Tue Jul 21 2009, 09:47AM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

In all the laser/TEC drivers I've designed, I've only ever used what you called the single term beta equation.

I also showed that if you put a 10k @ 25'C NTC thermistor into a Wheatstone bridge with three more 10k resistors, then the transfer function from temperature to output voltage is a S-shaped curve, that you can approximate as a straight line with only a degree or two of error over the range 10 to 40'C. This was how the temperature readout on my standalone TEC controller worked. The PC-hosted ones used a more math-heavy approach, with the beta equation substituted into the potential divider equation and rearranged for T.

The "10k @ 25'C" NTC thermistor is a de facto industry standard, they all have roughly the same A and B coefficients.

GeordieBoy

Tue Jul 21 2009, 11:38AM

Registered Member #1232 Joined: Wed Jan 16 2008, 10:53PM
Location: Doon tha Toon!
Posts: 881

> ...a S-shaped curve, that you can approximate as a straight line with only a degree or two of error over the range 10 to 40'C

That is exactly what I have done in the past too for things like battery chargers where the float voltage needs to be temperature compensated. The shallow S-shape complies well with the required characteristic over a given range of temperatures and then curves away dramatically outside of this range!

Back in the days of analogue synthesisers people were connecting thermistors with fixed resistors in weird series/parallel arrangements in attempts to "linearise" their behaviour over the range of typical room temperatures.

As a side note, regarding software and lookup tables: It is sometimes easier to turn the lookup table around and have an entry for every 1'C telling you what ADC result corresponds to this actual temperature. This is often more efficient than having a LUT entry for every possible ADC result, especially if you only need to know the final temperature to something like 1 degree accuracy. All you do is compare the ADC result with the LUT entries until you find which one it is closest too.

-Richie,

mikeselectricstuff

Tue Jul 21 2009, 10:53PM

Registered Member #311 Joined: Sun Mar 12 2006, 08:28PM
Location:
Posts: 253

GeordieBoy wrote ...

As a side note, regarding software and lookup tables: It is sometimes easier to turn the lookup table around and have an entry for every 1'C telling you what ADC result corresponds to this actual temperature. This is often more efficient than having a LUT entry for every possible ADC result, especially if you only need to know the final temperature to something like 1 degree accuracy. All you do is compare the ADC result with the LUT entries until you find which one it is closest too.

-Richie,

This is a useful way to save space, as you generally only care about a subset of the ADC range, anything else being 'too hot' or 'too cold'.

Another handy trick where you need a wide temp range from a thermistor is to have 2 pullups, switched on IO pins, so you switch in a lower value pullup & re-measure if the first reading gives a low resistance value.

Also bear in mind that table size can often be reduced significantly by a simple linear interpolation between table steps. This is especially so for PT100 linearisation - ISTR that 4 degree steps gives something like 0.01 C max error over most of the range, but then PT100s are rather closer to linear than thermistors.

tobias

Wed Jul 22 2009, 02:40AM

Registered Member #1956 Joined: Wed Feb 04 2009, 01:22PM
Location: Jersey City
Posts: 172

Why not to use a LM35?
0 C = 0 V
100 C = 1 V, linear. Easy to code.

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