Calibration equations and propagation of uncertainty
A calibration equation for a platinum resistance thermometer, for example, is determined from measurements of the sensor resistance at different temperatures. An important question is “how do uncertainties in the calibration measurements, resistance or temperature, affect the uncertainty in a measured temperature?”
The uncertainty propagation problem is important for temperature and humidity measurements because we rely so much on sensors and calibration equations.
In practice there are two classes of uncertainty propagation problems here:
- Interpolation: In this case the number of measurement points is exactly the same as the number of parameters in the calibration equation. This is usually not the preferred case because there is no redundancy in the measurements to allow us to make sure all measurements are self-consistent and that we have not made any mistakes.
- Least-squares fits: In this case we use many more measurements than the minimum required, and the equation is determined using a least-squares fit.
The propagation of uncertainty for the least-squares case is well known and there are many books giving the result. Surprisingly, the mathematics for the simpler interpolation case was not known. Although the interpolation case is not the preferred option, there are two important examples in thermometry where we have little choice, namely, standard platinum resistance thermometers and radiation thermometers. Once we solved this problem, we found that the mathematics explained how to calculate what happens when calibration equations are used twice in the same experiment – do error effects cancel or add? This has an impact on both humidity measurement and radiation thermometry.
- Interpolation with standard platinum resistance thermometers (SPRT)
- Interpolation with radiation thermometers
- Using a calibration equation twice
The ITS-90 specifies the mathematical relationship between measured SPRT resistance and temperature. This follows a three-step process:
(i) convert measured resistance to resistance ratio (to eliminate major differences between thermometers),
W = R(T)/R(0.01 ºC),
(ii) convert resistance ratio to reference resistance ratio (to eliminate most of the remaining differences between thermometers),
Wr = W – Δ(W),
where Δ(W) are the so called ‘deviation functions’ specified by ITS-90 for the various sub-ranges,
(iii) convert reference resistance ratio to temperature,
T90 = T90(Wr),
where the T90(Wr) function is fixed and defined by ITS-90. The second step is the most complicated because it is an interpolation. The figure below shows graphically the interpolation process for two SPRT calibrations over the water-aluminium sub-range (0.01 ºC to 660.323 ºC). Note the mapping from measured resistance ratio, W, to Wr. The solid curve of the figure passes through all four calibration points (water triple point, tin, zinc and aluminium freezing points). For the second curve (dotted), a small error has been introduced into the measured zinc-point value, WZn. The new curve passes through the new zinc point but continues to pass through the other, original, calibration points. The difference between the two curves shows how an error in the zinc-point measurement propagates to other Wr values and hence to temperatures in between the fixed points.
Caption: The influence of an error in a fixed-point measurement. The difference in the two curves is the fZn(W) sensitivity coefficient for the water-aluminium sub-range.
The difference between the two curves in the figure is called the fixed-point sensitivity coefficient, which we represent simply as fZn(W); the subscript identifies the fixed point, and the sub-range is usually apparent from the context.
In the case shown in the figure, we can very easily work out the equation for fZn(W) because it is a cubic equation passing through 1.0 at the zinc point and zero at the other fixed points:
It turns out that the original equation at step (ii) can be rewritten very easily in terms of such functions, and this is exactly the form for Lagrange interpolation. That is, by rewriting our interpolation equations in the form of Lagrange interpolations, we can readily identify the functions required to propagate the uncertainties in the measurements.
Unfortunately, only a few of the ITS-90 equations are simple polynomials that lead to Lagrange interpolations. However, they can all be written in a Lagrange-like form, so we can still identify the sensitivity coefficients. Although it looks like a minor breakthrough, this leads to a rather significant simplification in the uncertainty analysis for SPRT calibrations.
The propagation of uncertainty problem is even more complicated for radiation thermometers, because their response is a very non-linear function of temperature. There are a variety of different calibration methods for radiation thermometers, distinguished largely by the number of fixed points used. Many mid-temperature radiation thermometers are calibrated using the equation
where the values of A, B and C are determined from measurements at three fixed points. Because of the non-linearity of this equation, it cannot be expressed in the Lagrange-like form as for the SPRT equations. However, it turns out that the propagation of uncertainty equation has the same basic form for all interpolating equations – the sensitivity coefficients are just more difficult to calculate.
The calibration equation for radiation thermometers is very similar to Planck's blackbody radiation law. Peter Saunders of MSL managed to show how the A, B and C parameters can be determined from measurements of the spectral responsivity of the radiation thermometer. This was a major breakthrough because it made it possible to use the same equation for all of the different calibration methods for radiation thermometers, including ITS-90 and a novel two-fixed-point method developed by VSL (Netherlands - formerly NMi-VSL) and NMIJ/AIST (Japan), and enabled us to calculate how the thermometer responsivity affected the uncertainty propagation. Peter won the NZ Royal Society Cooper Medal for this work.
It is common in metrology to apply a correction equation several times during calibration and later when the thermometer is used. This occurs for example, when applying non-linearity corrections to our reference radiation thermometers. A correction is applied to each of the fixed-point measurements and again when the thermometer is used use to measure an unknown temperature. These corrections are highly correlated. How do the uncertainties in the non-linearity corrections propagate?
Very similar questions arise in many areas of thermometry, but the consequences are not usually very significant. For the radiation thermometers the uncertainties in the non-linearity corrections are quite small, and the mathematics follows directly from the propagation of uncertainty equations. However, the ‘correlation effect’ has an impact on humidity calibrations because of the way that gasses of known humidity are generated. Typically air is saturated with water vapour at a known temperature and pressure (so the relative humidity under those conditions is 100%), then the temperature and pressure are changed so that the vapour pressure of the water vapour changes to correspond to the required humidity under the new conditions. To calculate the dew-point temperature or relative humidity under the new conditions requires us to use twice the equation that relates the vapour pressure of water to temperature. If the initial and final conditions are very similar, then any errors in the vapour-pressure equation more or less cancel. If the initial and final conditions are very different, it is possible for the effects to add detrimentally. This is a major area of research for MSL at present.
For further information on radiation thermometers contact Peter Saunders
For further information on the humidity measurements contact Jeremy Lovell-Smith.
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