ECE 102 Engineering Computation

Transcription

1 ECE 102 Engineering Computation Phillip Wong Error Analysis Accuracy vs. Precision Significant Figures Systematic and Random Errors

2 Basic Error Analysis Physical measurements are never exact. Uncertainty in the values of measured quantities is due to errors in the measurement process. Errors can be reduced but not eliminated. Using measured values in a calculation will lead to uncertainty in the calculated result. How is accuracy related to uncertainty? 1

3 Accuracy versus Precision Accuracy is how near the mean measured value is to the correct or true value. Precision is the measurement repeatability, i.e., the distribution of the measured values. Not Accurate Not Precise Not Accurate Precise Accurate Not Precise Accurate Precise 2

4 Quantifying Accuracy Let x m be the measured value and x ref be the reference ( correct or true ) value. Absolute: abs = x m x ref Relative: rel = x x m x ref ref Scaled to x ref These equations assume you know what the true value should be. 3

5 Example:(Absolute vs. Relative Difference) x m = mm, x ref = mm Δ abs = mm, Δ rel = 10% x m = mm, x ref = mm Δ abs = mm, Δ rel 0.1% x m = 1.1 km, x ref = 1.0 km Δ abs = 0.1 km, Δ rel = 10% 4

6 Ways to Express Uncertainty Measured values should include information describing the accuracy of the measurement: Method 1: mean value ±uncertainty ( x ± δ x ) This specifies a range that the actual value is likely to fall within. Method 2: Significant figures A significant figure is any digit used in writing a number that is considered meaningful or reliable as a result of a measurement or calculation. 5

7 Method 1: mean value ±uncertainty ( x ± δ x ) Suppose the same measurement is performed repeatedly. mean x average of the measurements uncertainty δ x Maximum error δ x = (x max x min ) / 2 Probable error P(x min x x max ) = 0.5 Standard deviation δ x kσ, where k= 1, 2, or 3 (assuming error distribution is Gaussian) 6

8 Method 2: Significant Figures When reading instruments, the last digit is normally an estimate. It is standard practice to count one doubtful digit as significant. Example: If the measured value is 3.15, then there are three significant figures. The digit 5 is considered significant, but doubtful. The actual value may fall within the range v <

9 Rules for Interpreting Significant Figures Note: The significant digits are underlined in the following examples for emphasis. All non-zero digits are significant. Example: 8.936, 1.232, All zeroes located between significant figures are significant. Example:2501, ,

10 For integer numbers greater than one, all zeroes placed after the significant figures are not significant. Example: 25010, 25000, If a decimal point is used after an integer number larger than one, the zeros preceding the decimal point are significant. Example: , , Notice the decimal point 9

11 Zeroes placed after a decimal point that are not necessary to set the decimal point are significant. Example:359.00, 1.300, 0.10 For numbers smaller than one, all zeroes placed before the significant figures are not significant. Example: , ,

12 Numbers derived from pure mathematics (i.e., non-measured values) have effectively an infinite number of significant figures. Example: π ein e x 4/3and 3 in V = 4/3 π R 3 Non-measured numerical values have no effect on the significant figures of a calculated result. 11

13 Example: Quantity # sig figs

14 Rounding Methods Round towards nearest (most common) Increase the last digit retained by 1 if the first figure dropped is 5. Example: Nearest 0.1: , Nearest 1: , Nearest 10: , Nearest 100: , Alternatives: Round towards zero or +/- infinity 13

15 Preserving significant figures during arithmetic: Multiplication and division The product or quotient should contain the same number of significant figures as those contained in the number with the fewest significant figures. 14

16 Example:(assume values are measured) ( 1.0)( 0.01) = 0.01 ( 2.5)( 0.05) ( 1.0)( 01. ) = 0.1 ( 2.5)( 0.5) ( 1.0)( 0. 10) = 0.10 ( 2.5)( 0.50) ( 1.0)( 1) ( 1.0)( 1.0) ( 1.0)( 1.00) ( 1.00)( 1.00) ( 1.0)( 10) = 1 = 1.0 = 1.0 = 1.00 = 10 = ( 2.5)( 5) = ( 2.5)( 5.0) ( 2.5)( 5.00) ( 2.50)( 5.00) ( 2.5)( 50) = = = = 1 10 = = = 12.5 = = 1 10 ( )( ) = 10. = ( )( ) = =

17 Exact conversion factors do not affect the number of significant figures in a calculation. Non-exact conversion factors do affect sig figs. Example: 2.54 cm 1in 2 ( 72 in) cm 2.54 cm 1in 2 ( in) cm In this case, the measured value (and not the exact conversion factor) controls the sig figs. 2.5 cm 1in 2.5 cm in 1in 2 ( 72 in) cm 2 ( ) cm Here the conversion factor was rounded off, so it does affect the number of sig figs. 16

18 Addition and subtraction The answer should show significant figures only as far to the right as is seen in the least accurate number in the calculation. Example: x.xx y.yyyy + zz.zzz Least accurate, so it controls sig figs of result 17

19 Example:(assume values are measured)

20 Combined operations If products or quotients are added or subtracted: Perform the multiplication or division first Establish the correct number of significant figures in the sub-answer Perform the addition or subtraction Round to the proper significant figures Note: If intermediate rounding is not practical, then perform the entire calculation and set the proper significant figures afterward. 19

21 Example:(assume blue values are measured) y = (0.0250)(16.200) x + (2.14) x 2 for x = 3.2 Calculator result without using sig fig math: y = Calculation using the rules of sig fig math: + Perform multiplications first 1 st product:(0.0250)(16.200)(3.2) = nd product: (2.14)(3.2) 2 = Perform addition next is the result when sig fig math is utilized 20

22 Uncertainty in the Measurement Instrument A measurement instrument provides: The mechanical, electrical, chemical, etc. interface for performing the measurement A calibrated measurement standard The measurement accuracy of an instrument is limited by: Instrument design Operating & environmental conditions 21

23 Manufacturers specify instrument accuracy by: % reading EX: Spec = 1% reading, Read 5 V on 10 V scale Δ= ±(5 V)(0.01) = ±50 mv 5 V ±50 mv % full-scale reading EX: Spec = 1% FS reading, Read 5 V on 10 V scale Δ= ±(10 V)(0.01) = ±100 mv 5 V ±100 mv ppm(parts per million) of reading EX: Spec = 200 ppm, Read 5 V on 10 V scale Δ= ±(5 V)(200/10 6 ) = ±1 mv 5 V ±1 mv 22

24 Uncertainty in the Measurement Process Measurement errors fall into two categories: Systematic errors Random errors To achieve an accurate and precise measurement, both systematic and random errors must be accounted for. 23

25 Systematic Errors Systematic errors displace the mean value of a measurement in a consistent manner. Typical sources: Incorrectly calibrated or uncalibrated instruments Consistently improper use of an instrument Unaccounted for effects in the system May be reduced (but not eliminated) by applying a correction factor via a calibration process 24

26 Random Errors Random errors fluctuate from one measurement to the next and are distributed about a mean value. Typical sources: Instrument insensitivity Extraneous noise Intrinsic statistical process May be reduced (but not eliminated) by the averaging of repeated measurements. 25

27 Error Reduction Methods Calibrating the instruments Reduces the effect of systematic errors Increases the accuracy of the measurement Does notreduce the random error Averaging multiple measurements Reduces the effect of random errors Increases the precision of the measurement Does not reduce the systematic error For best results, use both methods. 26

28 Example: Vector Network Analyzer Electronic Calibration Module Mechanical Calibration Kit 27