UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 1 Uncertainty Analysis Now we will use what we learned in Chap. 4 to estimate the uncertainty of actual measurements. Remember that errors can be divided into two categories, bias and precision errors. The true value of a quantity is related to the mean of several measurements by: ( %)xx x U P xx = x (P%) (4.1)UUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 2 Zero Order Uncertainty All errors except instrument resolution are perfectly controlled. 01resolution (95%)2u Instrument uncertainty, uc, is an estimate of the systemic error.UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 3 Combining Elemental Errors: RSS Method 22 2 2x 1 2 kK2jj=1= + + +u e e e (5. )= (P%)e As a general rule P = 95% is used throughout all uncertainty calculations. Remember ±2δ accounts for about 95% of a normally distributed data set! Design-Stage Analysis Ex: Spa temperature regulation using a 3 digit voltmeter and thermocouple. Ex: What is the smallest zero-order uncertainty, u0, obtainable with the ADC used in our lab?UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 4 Error Sources Errors can arise from three sources: Calibration Data Acquisition Data Reduction TABLE 5.1 Calibration Error Source Element (j) Error Sourcea 1 2 3 4 5 Etc. Primary to interlab standard Interlab to transfer standard Transfer to lab standard Lab standard to measurement system Calibration technique aBias and/or precision in each element.UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 5 TABLE 5.2 Data Acquisition Error Source Group Element Error Sourcea 1 Measurement system operating conditions 2 Sensor-transducer stage (instrument error) 3 Signal conditioning state (instrument error) 4 Output stage (instrument error) 5 Process operating conditions 6 Sensor installation effects 7 Environmental effects 8 Spatial variation error 9 Temporal variation errer Etc. aBias and/or precision in each elementUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 6 TABLE 5.3 Data Reduction Error Source Group Element (j) Error Sourcea 1 Calibration curve fit 2 Truncation error Etc. aBias and/or precision in each element.UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 7 Error Propagation Most measurements are subject to more than one type of error. We need to estimate the cumulative effect of these errors. It is unlikely that all of the errors will be in one direction - more likely there will be some cancellation. The root-sum-squares (RSS) approximation is a good estimate: Since the overall result may be more sensitive to some errors than to others, we need to consider the functional relationships between the output and the various inputs. 2 2 2x 1 2 kK2jj=1 = + + + U e e e (5.1)= (P%)eUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 8 Error Propagation Continued The uncertainty in the dependent variable will be related to the uncertainty in the independent variable by the slope of the curve. (5.2)(5.5 in 2nd Edition)yxx=xdy=uudx If we have more than one independent variableUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 9 The true mean R' can be obtained from the sample mean Rwith a precision ± uR where and In order to account for the different sensitivities of the measurement to different inputs, we define a sensitivity index: and thus (5.3)(5.6 in 2nd Edition)1 2 L1R = { , ,..., }fx x x (5.4)(5.7 in 2nd Edition)RR = R (P%)u (5.5)(5.8 in 2nd Edition)1 2 L1R = { , ,.... } fx x x (5.6)(5.9 in 2nd Edition)1 2 LRx x x2 = { , ,.... }fu u u u (5.7)(5.10 in 2nd Edition)iix=xiR = i = 1,2,....,L |xUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 10 (5.8)(5.11 in_ 2nd Edition)iL2Rixi=1 = ( (P%))uuUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 11 Design Stage Example Example 5.3 A voltmeter is to be used to measure the output from a pressure transducer that outputs an electrical signal. The nominal pressure expected will be about 3 psi (3 lb/in.2). Estimate the design-stage uncertainty in this combination. The following information is available: Voltmeter Resolution: 10 μV Accuracy: within 0.001% of reading Transducer Range: ± 5 psi Sensitivity: 1 V/psi Input power: 10 VDC ± 1% Output: ± 5 V Linearity: within 2.5 mV/psi over range Repeatability: within 2 mV/psi over range Resolution: negligible KNOWN Instrument specification Assumptions Values representative of instrument at 95% probability Find uo for each device and ud for the measurement systemUncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 12 Design Stage Example Solution The procedure in Figure 5.3 will be used for both instruments to estimate the design stage uncertainty in each. The resulting uncertainties will then be combined using the RSS approximation in estimate the system ud. The uncertainty in the voltmeter at the design stage is given by equation 5.10 as (5.17 in the second edition) From the information available, For a nominal pressure of 3 psi, we expect to measure an output of 3V. Then so that the design-stage uncertainty in the voltmeter is 22d o cE E E( = ( + ( ) ) )u u u oE( = 5 V (95%))u %cE(u = (3 V X 0.001 ) = 30 V (95% assumed))UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 13 The uncertainty in the pressure transducer output at the design stage is also given by (5.10). Assuming that we operate within the input power range specified, the instrument output uncertainty can be estimated by considering each of the instrument elemental errors of linearity, e1, and repeatability, e2: Since (Uo) negligable, (0) V/psi, then the design-stage uncertainty in pressure in terms of indicated voltage is (ud)p = ±9.61 mV (95%). But since the sensitivity is 1 V/psi, this uncertainty can be stated as (ud)p = ±0.0096 psi (95%). 22dE(u = + =30.4 V (95%))5 30 22c 1 2p22( = + (95%assumed))u e e= +((2.5mV/psi x 3psi) 2mV/psi x 3psi)= 9.61mV(95%)UncertaintyAnalysis.doc 11/7/2007 1:11 PM Page 14 Finally, ud for the combined system is found by use of the RSS method using the design-stage uncertainties of the two devices. The design-stage uncertainty in pressure as indicated by this measurement system is estimated to be 22d d dEP22 = ( + ())U U U = 0.030mV +(9.61mV (95%))) = 10.06mV(95%) or 0.010