Math question: usage of relative coefficients

Factual Questions
1

Let’s say a measurement instrument has a temperature coefficient of +2% / °C, with 25 °C being the reference temperature. This can be anything; a gage block, voltmeter, strain gage, etc. (I understand +2% / °C is unrealistically high for most things, but this is just an example. Besides, I’m actually calibrating something that has such a value…)

So here’s my question: what does +2% / °C mean?? I can think of a few different ways of using it.

Here’s an example: Let’s say I have an instrument that reads “123” at 25 °C, with a temperature coefficient of +2% / °C. The temperature then shifts to 30 °C. What’s the new reading?

Here’s one way to calculate it:

The temperature has changed by +5 °C. This means the reading should change by 52% = 10%. The new reading is thus 123 + 0.1123 = 135.3.

Here’s another way to do it:

When the temperature is 25 °C, the value is 123.
When the temperature is 26 °C, the value is 123 + 1230.02 = 125.46.
When the temperature is 27 °C, the value is 125.46 + 125.460.02 = 127.97.
When the temperature is 28 °C, the value is 127.97 + 127.970.02 = 130.53.
When the temperature is 29 °C, the value is 130.53+ 130.530.02 = 133.14.
When the temperature is 30 °C, the value is 133.14+ 133.14*0.02 = 135.80.

Or what if I use a continuous compound formula such as what’s used to compute $$ interest, i.e.

New value = 123 * e[sup]0.02*(30-25)[/sup] = 135.94.

Now I understand there’s not a whole lot of difference between the three answers. But dammit, I wanna know the right way to do it! The other engineers around my workplace just shrug their shoulders.

So again, how is someone supposed to use a temperature coefficient spec such as +2% / °C?

2

I certainly wouldn’t be surprised if NIST or someone like that has an established policy, but I think it probably depends on the device & its manufacturer.

I’d use the last method, since although it assumes the same rate of change, it gives the best answer using that rate of change. Practical matters aside, this is the only way to use the spec as indicated – it gives the answer using a ‘perfect’ device that really does change by 2%/deg C over the whole temperature range.

From a practical point of view, whatever spec they came up with has to be dependent on their testing method. Unless it’s specifically designed to have the same characteristic over some range, the 2%/deg C may be a poor approximation for something that isn’t terribly exponential anyway. My guess is that they most likely determined it as the least error approximation over the whole range (at least that’s how I’ve done that sort of thing before).
If it is claimed to have the same characteristic, you’d think they’d give you a curve or something. Did they even give you a temperature range for which it’s valid? If not, it might only be valid in a fairly narrow range near 25 C.
(Again, one would hope they tell you that, but they don’t always.)

3

alpha - Temperature Coefficient
Ts,Ms - Reference Temperature, Measurement
T - Current Temperature
M - Measured Variable

Then if its linear scale,

M = Ms +alpha*(T-Ts)

Look at the specs of the instrument. It is usually linear, if not the the manufactured will specify.

4

It depends on what you’re assuming is constant. The value +2% / °C that you’re given only provides you with the slope of the thermal-change curve (explicitly, (dx/dT)/x, where x is the parameter whose temperature coefficient you’re given) at a single temperature. If you assume that this fractional expansion coefficient is constant with temperature, then your compounding formula is correct. But maybe dx/dT is nearly constant; in this case your first method is correct. Or maybe neither is constant… Basically, you need more information to know how to correctly extrapolate beyond linear order. Depending on what this is a thermal coefficient of, you might be able to use some knowledge of how such things typically change with temperature to determine what’s actually constant in your case.

5

Just a follow-up with a couple examples: Here (PDF) is a short introduction to the theory of thermal expansion in solids. This has a graph showing a representative thermal expansion coefficient. The coefficient is nearly constant for high temperatures (so the compounding formula is a good approximation), but for temperatures around room temperature or below it’s no longer such a good approximation. There’s also an example of a material with negative thermal expansion coefficient, which will require a more complicated model.

Thermistors (components designed to have resistance changing rapidly with temperature, and often used in temperature sensors) can have either negative or positive temperature coefficients. Here (another PDF) are some application notes for NTC thermistors, giving some representative graphs and fitting equations.

There are lots of other sources of temperature dependence. Once you know the source of the temperature dependence you can probably find some graphs showing generic behavior and use a fit based on those to decide how to extrapolate.