I am looking for equations that solve for temperature in terms of RTD resistance. I find lots of references to equations that only include the lower order terms or that solve for resistance.

I am writing firmware for a temperature measurement instrument. The instrument will work with 100pt, 1000pt, 120ni, and 10cu type RTDs and cover below zero ranges. I have found second order equations that solve for temperature for Platinum and Nickel. I have found just a first order equation for Copper. It looks like I need to include higher order terms for accuracy below zero.

Here are a few comments based on my experience as a mathematically inclined hobbyist, but certainly not an electronics professional.

In the past, I looked up some information about negative temperature coefficient thermistors, and came across an article containing some equations for resistance versus temperature.

I was disappointed from a purely scientific viewpoint when I realized that the equations, tagged with the names of people who developed or promoted them, were simply the result of curve fitting some measured calibration data to, typically, a third degree polynomial. There is NO scientific content in the derivation of those equations, at least from what I saw.

The equations are simply ad hoc, ex post facto, or purely empirical (take your pick of adjectival phrases).

The article,

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that John Larkin mentioned is quite in the same vein as the thermistor article that I came across, except that the Callendar-Van_Dusen_equation article is a mere stub.

John's comment "We usually use lookup tables and interpolation" seems to me to be at least as good an "explanation" as any of the equations.

Welcome to the real world. Most transmission-line impedance equations are the same, accidental coincidences of shape that were tweaked to work. It's just like fitting a polynomial to experimental data, but using any old equation that looks good. Such fits tend to produce insane results out of their sweet spots, like negative impedances for wide pcb traces.

Incidentally, there are NIST polynomials for thermocouple voltages, up to something absurd like 14th order, also probably free of deep theory.

I doubt that theory exists to calculate RTD or thermocouple R:T curves from first principles, to millikelvin accuracy. And don't even think about thermistors.

It's a lot faster to compute. I usually work in bare-metal assembly language, so interpolation is easier than high-order polynomial expansion or whatever. The tables can be generated directly from the NIST polynomials with a little BASIC program, and the resulting source file tossed into the assembly program.

I'm an engineer. I don't have to understand it, I only have to make it work.

I did these:

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This one also does cryogenic diode sensors, where there's even less theory to explain what's going on:

The Drude theory of metals says that a pure (non-alloy) metal has resistance proportional to Kelvin temperature. It works for copper, platinum, and lead pretty well (Heike Kammerling-Ohnes found something surprising at very low temperatures for lead, though). Alas, lead and copper tarnish (the calibration would shift, which is ... inconvenient), and most thin-film resistors have strains which arise from mismatched thermal expansion with the substrate. Strain is just like an impurity, it invalidates the Drude model. All real metals have some strains when you bend/wind/plate resistors out of them...

Yes, that's right. Thermocouples are designed around alloys (like constantan) that were commercially available in highly controlled prescriptions, like for low-tempco resistance wire. As long as the alloy is EXACTLY prescribed, the thermocouple is repeatable. I've used thermocouples where 0.07% of iron content was the difference between the (+) and (-) thermocouple wire; that has to be 0.07% plus/minus .0001% or so.

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