Uncertainty of a PT100 temperature calibration

Question

Description of the Problem:

I have a PT100 that is calibrated by an external lab and used in our lab as a reference for calibrating thermocouples. In the end, I would like to have a complete uncertainty budget for temperatures measured with the calibrated thermocouples.

I would really appreciate any help in the calculation of the uncertainties. I'll describe my thought process below.

What I did so far:

The PT100 was sent to a lab to calibrate and I got back a table that contains three columns of data:

1. the true temperature $T_\mathrm{90}$ as defined in the ITS-90
2. the temperature $T$ measured with the PT100
3. the measurement uncertainty $u(T)$ from the temperature measurement with the PT100

From this data I learn that the temperature measured with the PT100 is too low, which gets worse with higher temperature. For example, at 300 °C, the PT100 measures 299,5 °C and at 500 °C, the PT100 measures 498 °C (values are rounded, in the sheet from the lab all values are given with a scale of two).

I would like to have a function that corrects for this error and did the following (with OriginLab 2021):

1. Plot $T$ versus $T_\mathrm{90}$ whereas I put $T$ on the x-axis (thought it would be easier to fit it with a quadratic function that way, see step 2)
2. Fit the data with a function in the form of $y = y_0 + aT + bT^2$, which gives me the coefficients $y_0$, $a$ and $b$ together with their uncertainties

Now in the function $y$ I can enter the temperature measured with the PT100 and get the corrected value. For the calculation of the uncertainty $u(y)$ I did the following, using the laws of error propagation:

3. Calculate the derivatives $\partial_\mathrm{y_0}y=1$, $\partial_\mathrm{a}y=T$, $\partial_\mathrm{b}y=T^2$ and $\partial_\mathrm{T}y=a+2bT$ (with $\partial_\mathrm{x}y=\frac{\partial y}{\partial x}$)
4. Calculate the uncertainty $u(y)$ as $u(y)=\sqrt{u(y_0)^2+[T u(a)]^2+[T^2 u(b)]^2+[(a + 2bT) u(T)]^2}$

What I got:

Here is a plot of the temperature difference from the uncorrected temperature of the PT100 and the corrected temperature of the PT100 to the temperature $T_\mathrm{90}$, together with the uncertainties $u(T)$ and $u(y)$:

Now my final questions are:

1. Is this approach correct? Because later I would like to use $y$ and $u(y)$ instead of $T_\mathrm{90}$ to perform a similar correction for the thermocouples.
2. Is it correct that the error of $y$ rises with the temperature? (Why is it not constant?)

Edit after reading @Urb's answer:

Origin shows $R^2$ as 1.

To reduce the uncertainties I thought: What if I try to fit the data with $y = y_0 + bT^2$, leaving out the $aT$ term? Since this didnt really work I fitted the difference $T_\mathrm{90}-T$ vs. $T$. The error is $u(y)=\sqrt{u(y_0)^2+[T^2 u(b)]^2+[2bT\cdot u(T)]^2}$.

The corrected temperature is then $T_c=T + y$ and the uncertainty is $u(T_c)=\sqrt{u(T)^2+u(y)^2}$. If I plot the difference $T_\mathrm{90}-T_c$ vs. $T$ I see that the uncertainties are much smaller, even in the range of $u(T)$ (but also growing with temperature, which I understood now why). Does this calculation make any sense?

Additional question: In this scenario $T_\mathrm{90}$ had no uncertainty. But what do I do if it would have an uncertainty? How do I incorporate this in my calculations?
--> Saw that its the same as above with $a=1$... However, the uncertainty seems to be way lower...

Answer 1

1. The approach looks valid to me, given that the purpose is to obtain a better estimation of the actual temperature, which the model clearly does. I'm guessing that you chose the quadratic function by looking at the plot. Although there are only six points, you could do some checks: is the coefficient of determination $R^2$ high? Are the coefficients of the regression statistically significant doing a $t$-test? Is the dependence really quadratic? (all this seems ok given that none of the residuals $T_{90}-y$ in the figure look large compared with the others). Assuming the fit was done by OLS regression, one could still check if more assumptions hold: normality of the residuals, homoscedasticity, etc. but failure to accomplish any of these won't bias the coefficients, so the estimation is still good; just make sure not to predict the temperature for new values outside the range of temperatures (I'm guessing $[\sim 20,500]\ ^\circ \rm C$) where the fit was done.
2. The error of $y$ rising with temperature seems correct. Since the functional form of $y(T)$ is quadratic in $T$, for a fixed error $u(T)$ (which is what it seems from the errors in the black curve) the error in $y$ gets inflated as $T$ grows. A simplistic analysis shows that the interval $\left[y(T-u(T)),\ y(T+u(T)))\right]$ grows linearly with $T$.