EE 521: Instrumentation and Measurements

Transcription

1 Aly El-Osery Electrical Engineering Department, New Mexico Tech Socorro, New Mexico, USA August 30, / 19

2 1 Course Overview 2 Measurement System 3 Noise 4 Error Analysis 2 / 19

3 Textbooks C.W. De Silva, Sensors and Actuators: Control System Instrumentation, CRC Press, R. Northrop, Introduction to Instrumentation and Measurements, 2nd Edition, CRC Press, / 19

4 Topics Covered Measurement systems Error Analysis Signal conditioning Noise and interference Survey of different types of sensor input mechanisms Sensor Applications Data acquisition and digital signal processing 4 / 19

5 Grading Homework: 30% Final: 20% Project and Research Paper: 20% Presentations: 20% Class Participation: 10% 5 / 19

6 Scope Physical Quanitiy Sensor Amp QUM S + n 2 x + n 1 ADC LPF 6 / 19

7 Noise Sources 1 Environment noise: associated with the physical quantity being measured. 2 Measurement noise: due to the measurement system. 3 Quantization noise: due to digitizing the analog signal. 7 / 19

8 Error in Measurement 1 Gross Errors Taking measurement during transient time of the instrument. Mistakes recording data or calculating a derived measurand. Misuse of the instrument. 2 System Errors Calibration errors. Random noise both internal and external. Drift. 8 / 19

9 Error It is important to find a way to describe errors in measurements in terms of some accepted concepts. For example, accuracy, precision, limiting error and error statistics. 9 / 19

10 Accuracy Error ǫ n X n Y n (1) and %ǫ ǫ n Y n 100 (2) where Y n is the true value and X n is the actual measurement. 10 / 19

11 Accuracy Error and ǫ n X n Y n (1) %ǫ ǫ n Y n 100 (2) where Y n is the true value and X n is the actual measurement. Dilemma To compute the error we need to know the true value. If the true value is known, why measure it? 10 / 19

12 Accuracy or A n 1 Y n X n Y n (3) %A n 100 %ǫ (4) 11 / 19

13 Precision where P n 1 X n X X X 1 N (5) N X n (6) n=1 12 / 19

14 Deviation Deviation d n X n X (7) Average Deviation Standard Deviation D N = 1 N N d n (8) n=1 S N = 1 N N dn 2 = σ x (9) n=1 Variance S 2 N = 1 N N Xn 2 ( X) 2 = σx 2 (10) n=1 13 / 19

15 Limiting Error (LE) A term frequently used by manufacturers to describe worst case error Taylor s Series f(x ± X) = f(x) + df X dx 1! + + d n 1 f ( X) n 1 dx n 1 (n 1)! 14 / 19

16 Limiting Error (LE) Assuming that QUM is a function of N variables Q = f(x 1, X 2,...,X N ) (11) but due to measurement errors ˆQ = f(x 1 ± X 1, X 2 ± X 2,...,X n ± X N ) (12) Using Taylor series and ignoring the higher order terms. Q MAX = Q ˆQ = N f X j X (13) j j=1 15 / 19

17 Linear Regression - Least Mean Square Linear Fitting Fit a line with equation y = mx + b to a set of N noisy measurements. Minimize σy 2 = 1 N [(mx k + b) Y k ] 2 (14) N k=1 16 / 19

18 Linear Regression - Least Mean Square Linear Fitting Differentiating with respect to m and equating to zero σ 2 y m = 2 N N [(mx k + b) Y k ] X k = 0 (15) k=1 Differentiating with respect to b and equating to zero σ 2 y b = 2 N N [(mx k + b) Y k ] = 0 (16) k=1 17 / 19

19 Linear Regression - Least Mean Square Linear Fitting Solving the previous two equations of m and b m = R xy(0) Ȳ X σ 2 x (17) b = Ȳ X 2 XR xy (0) σ 2 x (18) where R xy is the cross-correlation of X and Y evaluated at zero R xy (0) = 1 N N X k Y k (19) k=1 18 / 19

20 Linear Regression - Goodness of fit Using the correlation coefficient Perfect fit if r = 1. r R xy(0) XȲ σ x σ y, 0 r 1 (20) 19 / 19