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:
- the true temperature $T_\mathrm{90}$ as defined in the ITS-90
- the temperature $T$ measured with the PT100
- 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):
- 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)
- 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:
- 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 $\partial_\mathrm{x}y=\frac{\partial y}{\partial x}$)
- 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:
- 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.
- 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...