Quantities Measurement Multivariable measurements 2 Measured quantities Leslie
Quantities & Measurement Multivariable measurements 2. Measured quantities © Leslie Pendrill 060606 Regression Sought quantities 1
Advanced Measurement Uncertainty Analysis Multivariable measurements 2. Regression y = A. x Least-squares fit: Variance-covariance: © Leslie Pendrill 060606 2
Measurement instruments 2. 1 The e. m. f. at a thermocouple junction is 645(10) µV at the steam point, 3375(3) µV at the zinc point and 9149(49) µV at the silver point. Given that the e. m. f. temperature relationship is of the form: E(T) = a 1 T + a 2 T 2 + a 3 T 3 (T in °C), cold-point at ice (Eice = 0(10) µV) find a 1, a 2 and a 3. © Leslie Pendrill 060606 Bentley exercise 2. 1 3
Advanced Measurement Uncertainty Analysis 2. 1 Answer E(T) = a 1 T + a 2 T 2 + a 3 T 3 Emf (V) T (C) Least-squares fit a 1 = 5. 840 µV/ °C © Leslie Pendrill 060606 a 2 = 6. 37 10 -3µV/ °C a 3 = -2. 651 10 -6 µV/ °C Bentley exercise 2. 1 4
Advanced Measurement Uncertainty Analysis 2. 1 Answer E(T) = a 1 T + a 2 T 2 + a 3 T 3 y = A. x Residuals M=4 a 1 = 5. 83969718 (0) µV/ °C N=3 a 2 = 6. 36808419 (0) 10 -3µV/ °C a 3 = -2. 65055967 (0) 10 -6 µV/ °C © Leslie Pendrill 060606 Bentley exercise 2. 1 5
Advanced Measurement Uncertainty Analysis Multivariable measurements 2. Regression d = A. c c= (AT. W. A)-1. AT. W. d Weighting matrix Measured quantities © Leslie Pendrill 060606 Sought quantities 6
Advanced Measurement Uncertainty Analysis Multivariable measurements 2. d = A. c Regression c= (AT. W. A)-1. AT. W. d Group variance © Leslie Pendrill 060606 Experimental variance 7
Advanced Measurement Uncertainty Analysis 2. 1 Answer E(T) = a 1 T + a 2 T 2 + a 3 T 3 Weighted least-squares fit xfit= (AT. W. A)-1. AT. W. y M=4 a 1 = 5. 8(4) µV/ °C N=3 a 2 = 6. 3(1. 5) 10 -3µV/ °C a 3 = -2. 6 (1. 2) 10 -6 µV/ °C © Leslie Pendrill 060606 Bentley exercise 2. 1 8
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