Calibration of the calorimeter
 

 The rtds will be calibrated to measure a difference of temperature and not a single temperature

 
The formula to find the true temperature needs the temperature in °C or K (nevermind which one since the physical object measured is a rise of temperature)
The error under 5% is for a rise of temperature bigger than 10°C ; Never measure a rise of temperature smaller than 1.5°C with those rtds, that's the limit to reach the 1% error


Calibration in air (aluminum)
   Problem of accuracy
    Coefficients found in air
    Why we can't know wether the fit is quadratic or linear
Calibration in vacuum (tungsten)
   Rtd offset
    Uncertainty calculation
    Final formula
   



The 2 big unknowns of the calorimeter are the rise of temperature in the Slug during a beam exposure and the heat capacity of this slug.
The temperature should be measured with 6 RTDs, whom datas will be analyzed by S518 board and then monitored on computer.
Nevertheless, due to the way they have been built, each RTD can display a temperature slightly different (up to 1K) from the others.
The calibration of the calorimeter is actually the calibration of those RTDs.

The first step to calibrate the RTDs is to define a temperature of reference. The reference Ω ("true temperature") will be taken by a pre-calibrated and highly accurate probe Omega DP25).
Six corrections have then to be found to make the temperatures given by the 6 RTDs agree with the temperature of Ω.
Those corrections should be known as functions, to be more precise :  polynomial (please have a look at the part why polynomial functions)

The first attempt to obtain a good calibration of the rtds has been made with the Aluminum slug, but the important calibration must be made withe the tungsten slug. The procedure should be exactly the same, i.e. heat the slug and measure the temperature with each rtd, and compare the results with Ω.
The rtds are placed  in the slug as follow :
 

calibration with the slug of aluminum in air :

This first calibration is only a "test", to find whether we can complete an accurate calibration in air.
Let's heat the slug up to approximately 75°C, and measure the evolution (decrease) of the temperature with the 6 rtds and the probe omega.


    It's obvious than the temperature measured by the rtds and the temperature Ω are close, but if we take a closer look (zoom below) we can notice there's a difference up to 1K.
Since we need a accuracy of 12mK (the better we can have, limited by a bit), theres' a lot of work...

Roughly 25 sets of datas are analyzed to obtain
the better accuracy.












It's not really easy to find the relation between each curves, so let's plot the temperature given by each rtd (call Trtd below) against Ω :


    You can see from the curve fit that the correlation coefficients for a linear relation between Trtd and Ω (calculated with the software Kgraph (software running only with Windows ;), using the relation ) are very good.

So we can try to plot the difference against Trtd and Ω against Trtd to obtain a better accuracy.


NB : why plot Trtd-Ω against Trtd instead of against Ω ?

    If we want to compare the temperature given by each rtd and see if it matches Ω, we need, for each rtd , obtain something like "Ω = g(Trtd)"
If we try to plot Trtd-Ω = f(Ω), we'll obtain Ω = Trtd-f(Ω), far less practical...




Now, let's have a look at the curves Trtd-Ω against Trtd :

    The symbol "*" written after Trtd near each axis means that the difference in the temperature given by rtd6 (Trtd6) and Ω is plotted against Trtd6 etc.
To make a fit of those curves, we'll remove the extremes, where noises could be observed.
We made a first and 2nd deg. polynomial fit of those curves.
Why a polynomial ? Because we intuitively reckons that the relation between the temperature given by a rtd and Ω . Of  course it's only an hypothesis and it should be wrong, but it's better to begin a fit with simple functions...
In fact, you will see below that  there are some problems with those fits (a sinus function, and problems to obtain good accuracy on coefficients).
On top of that, if you take a closer look at the documentation given with the rtd (see the glossary), you'll see that the fit should normally be made with Tchebychev polynomials between the resistance of the rtd and the resistance of the probe omega, but we'll suppose that the difference between the "real" function and the fit we tried is small, so we can forget Tchebychev.

Nevertheless, don't forget that the fit for the aluminum slug is only a preliminary test, so that it would be useless to spend to much time to have a high-precision fit, the fact that this calibration is in air is a big source of errors. If the tests in vacuum with the tungsten slug aren't satisfactory, we would change our methods, but now it's enough.

Anyway, here you can find the coefficients found for each set of curves for a linear fit, and here for the 2nd deg. polynomial. (format .txt)


Since we have a lot of coefficients, which one to choose to obtain the better function ?
We are confronted with a problem of accuracy :

Theoretically, we should obtain a pick for each coefficient, or at least a thin gaussan.


    But it's not exactly the case : first the width is bigger than what allowed by the error budget, and this histogram is a nice one... only a few of the coefficients found are  accurate enough to make that...

2 solutions : either the air is the biggest (and I hope only) problem, so that the measures would be far better in vacuum, or the 2 fits we tried aren't correct, so that we should try to found the function that could make an improved fit (very unpleasant solution).

In any case, that should wait the tungsten slug and the tests in vacuum to see whether it's better or not.












Using this method for each set of coefficients, we chose the following coefficients to fit the curves :

for the linear fit


NB : there should be a potential mix up in the correspondence between the name (P110~) and the number (0 to 7) of each rtd. We are waiting for more informations to sort out this problem (once more, in the case the runs with the tungsten slug are good, we could simply forget this part ;)


The biggest problem of this attempt to calibrate in air is the fact we can't know whether the fit is linear or quadratic (or something else, but it seems highly improbable) :  let's have a better look at the fit :  for two different sets of datas, let's plot the difference between the function we obtained with the linear fit and Trtd-Ω :


    Note that we have here only 2 fits obtained with 2 sets of datas, but the general shape of the others sets of datas is the same.

You can see that the difference between the curve and the fit is the same in the 2 cases, i.e. there's faint curvature that should be explained by a quadratic term, and a kind of sinus function.

so let's do the same with the 2-deg-polynomial fit.








    The curvature has disappeared, but the sinus function is still here. We hope this is due to noises, and that it should disappeared with the tests in vacuum, but maybe it's a remainder of the Tchebychev polynomials (we hope it isn't, because it would make the fit far more complicated).

On top of that, you can see that the temperature oscillate with an amplitude of roughly 200mK that we think due to the air too.







Now we have "chosen" the coefficients we can see if it fits with some datas taken in vacuum (with the aluminum slug) :

    You can see on the zoom, during an equilibrium phase, that the difference between the temperature measured by each rtd is up to 0.8K.

You can also see the temperature curves from rtd0 and rtd6 have the same shape, as well as the curves from rtd2 and rtd5.

Let's try to change the curves thanks to the coefficients obtained above :
















    The left picture below represents a zoom of the same curve modified with the linear coefficients, the right one with the 2nd deg. pol. (click to have a picture of the full curve).





















You can see there's an improvement, since the difference in temperature between the 2 extremal measurements is reduced to 0.159K and 0.161K (instead of 0.8K previously), but that's not enough.
Moreover, we still don't know yet which fit is the best (I've a weakness for the 2nd deg polynomial).
So let's wait for the measure with the tungsten slug in vacuum, maybe it would be better.


calibration with the slug of tungsten in vacuum :

The calibration in air isn't unusable (we will need it afterwards when measuring a rise of temperature ), but you can forgot the idea of calibrating the rtds to measure a single temperature.
In fact, the coefficients found in air disagree with the datas in vacuum : there's a difference of 0.2K, which is approximately the noise found in air : problem...
The second problem is it's difficult to put the probe omega in vacuum to made other runs and try to calibrate the rtd to give the true temperature (that's why we won't be able to do a accurate calibration for a single temperature).
Instead we can try to calibrate the rtds to give always the same temperatue, i.e. calibrate 5 rtds with a 6th one, so that we should be able afterwards to make all the rtd mesure the same rise of temperature.
So, let's determine the function linking 5 rtds with the rtd P11044 (named below 44 and the other 42, 47, 48, 51, 55).
Why  the rtd 44? because it works well with this one and that the accuracy is better (and the coefficient found for the tests in air are ones of the most accurate of the 6 rtds).

From the test runs in vacuum we have 3 series of 12 sets of datas like this one : serie jon, A & B. We'll work with the datas from jon and check the results with A and B.

Each set of datas is obtained by heating the slug (power of the heater variable to increase the range of intensity known), and by leting the system reaching an equilibrium (plateau after the pic of temperature).
The only interesting part of each data is this plateau :this is the only section where we are sure all the rtds are measuring the same temperature (when the equilibrium is reached in the slug).
So we can extract all the datas from those plateau and plot the temperature of a rtd against Trtd44.








We can suppose that we still have the problem we can't know if the rtds are linked with a linear or polynomial function.
But in fact the difference between the 2 functions is less than 6mK for the temperatures we considere, so it's not that important...
We'll considere that the rtds are linked between each other by the linear function, it's the simplest (and we'll show after it's highly more probable).
You can see from the linear fit that the slopes are close to 1, so let's make an hypothesis : the slope is 1 and the difference between the rtds is only a difference of temperature. (for the range of temperature considered 10-50°C)


To confirm (or refute) this hypothesis, let's plot the difference between the temperature given by an rtd and 44 :

You see that the difference between the rtds is a multiple of 12.5mK, that's the bit of a rtd : that confirms the hypothesis.
So we can considere now that the temperature given by each rtd differs from the temperature 44 by an offset multiple of  12.5mK :

 So we can say that we don't need the 12 functions (6quadratic and 6 linear) to measure a rise of temperature, but only the 2 for the rtd 44 (with the condition that the offset for each rtd doesn't change significantly, which is proven below with the calculation of the uncertainty and the negligible effect of the offset)




So we can now calculate the "true" difference in temperature ΔΩ and its uncertainty :

2 factors must be took into account to calculate this uncertainty :

    - As we can't know whether the true temperature is linked to T44 by a linear or a quadratic function (where T44 is the temperature given by the rtd 44 ), we must calculate the difference between ΔΩl and ΔΩq, rises of temperature obtained with the linear or quadratic function.
    - For each function (linear and quadratic) we must know the error due to the incertainty of the coefficients (that's why I chose the rtd 44, the uncertainty is the smallest).


with the coefficients found above, we can express Ωq = T44 - [M1*T44 + M2*(T44)² +M0] and  Ωl = T44 - [a*T44 + b]


and we also know that T44 = Tx + αx    where Tx  is the temperature given by the rtd "x" and αx  is the offset multiple of 12.5mK found above.

so we can express Ωq = Tx + αx -[M1*(Tx + αx) + M2*(Tx + αx)² +M0]                Ωl = a*(Tx + αx) + b

and the "true" difference ΔΩq = Ωf - Ωi = (1 - M1)*(Txf - Txi) - M2*(Txf² - Txi² +2*αx*(Txf - Txi))            ΔΩl =(1 - a)*(Txf - Txi)

So we have : ΔΩq - ΔΩl = (a - M1)*(Txf - Txi) - M2*(Txf² - Txi² +2*αx*(Txf - Txi)) = f(Txi,Txf)  the error for not knowing the fit. (and you can see that the effect of the offset is αx negligible ~5e-3%)

we can calculate the error due to the uncertainty of the coefficients with this formula :


So we know dΔΩq = q(Txi,Txf) incertainty for the quadratic fit and dΔΩl = l(Txi,Txf) for the linear fit. (Those functions are too awful to be written here, if you really want them, click here)

Since f(Txi,Txf)>q(Txi,Txf)>l(Txi,Txf), we can say that the trues rise of temperature is given by the linear fit (the simplest solution), with an error equal to  f(Txi,Txf)+q(Txi,Txf).      (if you disagree or don't understand, click here to have more explanations, word format)

We can now plot the function for the total uncertainty : t(Txi,Txf) = (f(Txi,Txf) + q(Txi,Txf) + l(Txi,Txf))/r(Txi, Txf) that gives us :
where r(Txi, Txf) is the rise of temperature with the linear or the quadratic fit



and you can show (for a rise of temperature bigger than 10°C) the relative error in the measurement of the difference in temperature is smaller than 4.8%, that is smaller than the error budget for the temperature (budget of 1%, reached for a rise in temperature bigger than 1.5°C).

So that's great ^^, we don't realy need to know whether th fit is quadratic or linear, 


The formula to measure a rise of temperature is :

0.9657*(Txf -Txi) +/- 0.5% (if Txf-Txi > 10°C)       where Txf is the final temperature and Txi the initial one, both measured with the same rtd.




Something hazy ? Need more explanations? contact yannick rousseau <rousseau@jlab.org> 
and sorry for all the grammatical mistakes :(