Hello:
I am trying to determine the uncertainty that the Logger commits on having provided values of temperature for two types of sensors. I have read in the forum on this topic but I continue having doubts. I write the cases for if someone can help me:
1) Vaisala: T=-40C+0,1(ºC/mV)*Vreading(mV)
DataLoger: CR1000
Conexión en Single Ended.
Input Range (mV)=2500 mV
CR1000 Specifications:
Accuracy (0C-40C): +/-0,06% of Redding + Offset
Offset: (3*667+3)*μV
If for example, Treading=0,054ºC then V=400,54 mV
Accuracy=+/-(0,06/100)*400,54 mV+(3*667+3)*1E-03 mV=+/- 2,244mV.
In degrees Celsius: Accuracy=+/- 2,244mV*0,1(oC/mV)=+/- 0,244 C
And associating a rectangular distribution , the incertidumbre is :
+/- (0,244 /(2*raiz(3)))= +/- 0,065ºC
Would not too big uncertainty be this? Can someone say to me if I am doing it correctly?
Which is the resolution of the DataLogger? I can see changes of temperature in 5 decimal numbers and if this one is 667μV =(667/1000)mV*0,1 (oC/mV)=0,0667ºC then they would be 4.
2) PRT 4-wire (BrHalf4W)
Vx = 2093 mV; Input Range =25 mV (Range 1 y Range 2); Input Reversal True
Instruction PRT() for conversion of resistance to temperature.
To determine the uncertainty have I to use the following expression?
Ratio Accuracy: (+/-0,04% of voltage Reading + Offset)/Vx
Offset: (1,5*667+1)*μV
How can I obtain the uncertainty in units of temperature?
Thank you very much
1)Regarding question 1.
Our datalogger accuracy specifications are conservative and so a uniform (rectangular) distribution as you are using seems appropriate. My understanding is that the standard deviation for a rectangular distribution equals a/square_root(3) = a/1.73 and that 1.65 standard deviations are required for 95% confidence for the rectangular distribution. So for a = 0.244C, a rectangular distribution would span from –0.244 to +0.244 and offer a standard deviation of 0.244/1.73 = 0.141 C. Then (1.65)*(0.141 C) = 0.232 C, resulting in a measurement uncertainty estimate of +/- 0.232 C for 95% confidence.
You could measure this Vaisala sensor differentially and use input reversal in order to reduce the offset term of our CR1000 accuracy specification, at the expense of a single-ended measurement channel. There is a subtly in the measurement error calculation for input reversal, and that is the fact that two independent measurements of opposite signs are then taken and added together and divided by 2 for the overall measurement result with input reversal. According to theory on the Propagation of Uncertainties, the overall estimated uncertainty of two numbers added is computed as the root-sum-square of the uncertainties of each measurement. Since the result is then divided by two, the measurement accuracy using a differential measurement with input reversal would be 1.414/2 = 0.707 times the measurement error for a single measurement.
Accuracy = +/-(0.707)*[0.0006*400.54 mV + (1.5*0.667 mV + 0.001 mV)] = +/-(0.707)*[0.240 mV + 1.00 mV] = +/- 0.877 mV which translates to +/- 0.088 C for the rectangular distribution boundaries, instead of +/- 0.244 C, which is helpful. I then get a resulting uncertainty estimate of +/- 0.084 C for 95% confidence with the differential measurement with input reversal.
Regarding your resolution question, a basic resolution of 667 mV for one single-ended measurement should result in 0.0667 C resolution with your sensor. So one would expect to see 0.0667 C toggle to 1.334 C based on measurement resolution for a single measurement. This resolution can be improved by averaging multiple measurements as long as there is dither between successive readings, which is why our specified differential measurement resolution is twice as good as compared to single ended measurements. Using the CRBasic averaging instruction will also enhance resolution, again as long as there is dither between successive measurements. Our 50 and 60 Hz rejection on the +/- 2500 mV and +/- 5000 mV input ranges use a multiple measurement approach which would enhance the single measurement resolution mentioned above. For a single-ended 50 Hz or 60 Hz rejection measurement on the +/-2500 mV or +/- 5000 mV input ranges, two measurements separated in time by 10 ms are averaged together for the final measured result. Differential 50 Hz or 60 Hz rejection measurements on the +/-2500 mV or +/- 5000 mV input ranges with input reversal take four total measurements, two for the first input polarity and two for the reversed input polarity. So I would expect you to see a 0.0667 C resolution for a single measurement, but improved resolution if multiple measurements are averaged.
Another possibility for enhanced resolution would be to use the CR1000 Autorange feature which performs a second measurement based on the best range determined from an initial measurement. Yet this would only enhance the resolution in your application at cold temperatures where the temperature sensor output was below 250 mV, which is the next step down from the +/-2500 mV input range.
2)Regarding question 2.
The expression (+/-0.04% of voltage Reading + Offset)/Vx is for a single measurement half bridge with excitation Vx. The BrHalf4W instruction is more complicated because it takes two differential measurements and ratios the results to arrive at the quantity we refer to as X. Because the measurements are ratioed, the improved ratiometric accuracy applies. This X is then used to solve for either the top or bottom resistor depending upon the bridge configuration. According to theory on the Propagation of Uncertainties to estimate the uncertainty of two quantities that are divided one needs to perform a root sum square of the RELATIVE uncertainties of numerator and denominator, where in this case each accuracy is computed from (+/-0.04% of voltage Reading + Offset). The relative uncertainty implies dividing the (+/-0.04% of voltage Reading + Offset) value by the appropriate Vsignal in each case instead of Vx, which then are to be combined in a root-sum-square fashion to get an accuracy estimate of the ratio X. I’m using the term accuracy as computed by our datalogger measurement accuracy specification as the quantity used to determine the limits of a rectangular distribution. Then an estimated uncertainty can be arrived at by the 1.65 X standard deviation method previously discussed using accuracy/square_root(3) = a/1.73 for standard deviation, which gives the desired uncertainty in X which I’ll term deltaX. The deltaX can be converted to a resistance uncertainty, although the conversion depends upon if the PRT is the top or bottom resistor of the half-bridge circuit. If the PRT is top resistor then deltaRtop = Rbottom*deltaX. Yet if the resistor is the bottom resistor of the half-bridge then deltaRbottom = -Rtop*deltaX/(X*X), which is arrived at by partial differentiation of the expression X = Rtop/Rbottom for the appropriate resistance. So finally we have deltaR, which can then be converted to deltaTemperature by using the ohm/C PRT temperature coefficient, such as 3.85 ohm/C for a 1000 ohm PRT. There is just one more addition I believe. You mentioned you were using input reversal, which from the previous example we know improves the offset, but results in two measurements which need to be dealt with by scaling by the factor 1.414/2 = 0.707 as we did in the first example prior to dividing the two measurements. I hope this helps. Sorry for the delay in answering.
One additional point in relation to your question (2). You quote the accuracy equation with the offset term in it, where you have used an offset based on a basic resolution figure of 667 uV.
If you are measuring on the 25mV range (for a PT100 as you imply) the basic resolution is 6.7 uV not 667.