INSTRUMENTS AND MEASUREMENTS SYSTEMS .

1. INSTRUMENTS AND MEASUREMENTS SYSTEMS.

2. OBJETIVES • Characterize the main elements that are involved in a Measurement Process. • Apply the standard proceedings to work with measurement data.

3. Lect. 1:INSTRUMENTS AND MEASUREMENTS SYSTEMS Sumary: • Measurement Process. • Characteristics of Instruments and Measurement systems. Static and Dynamic Characteristics. • Random Measurement Processing

4. REFERENCES

5. Measurement Concept • Comparison process Quantity (electrical signal or Physical variable) INSTRUMENT Measurement: Unknown magnitude XX.XX standard Units. Measurement=> How many times does the magnitude of the Quantity fit into the standard unit?

7. Characteristics of Instruments and Measurement systems • Static Characteristics: quantities are constant or have slow time variation. • Dynamic Characteristics: the quantities are changing with time. Under this condition the system is described by some differential equation.

8. Static Characteristics: • Range and Span. • Accuracy. • Precision. • Linearity. • Hysteresis • Resolution, threshold.

9. Error • No measurement is free from error. • Error can be related with the instrument property. It is given by the accuracy of the instrument. Depends on the calibration. • But error also can be related with the quantity which is measured or the process, it depends on the randomness of the process.

10. Absolute error We really don’t know the true value, we can estimate it with another instrument with better accuracy. If the input is random, the accuracy is limited by the accuracy of the instrument if we increase the number of measurements.

11. Relative error The same absolute error can be neglected or not according to the magnitude that is measured. R I=1A I=10A 10% 0.1% δI=0.1A

12. Range and Span • Range : interval between maximal and minimal reading  [xmax, xmin] • Span: difference between maximal and minimal reading. xmax-xmin Clinical thermometer: readings from 34 ̊C to 40 ̊C (range) , span= 6 ̊C

13. Error calibration Curve

14. Accuracy • Accuracy. It is the closeness with which an instrument reading approaches the true value of the quantity being ,measured . • Low absolute and relative value high accuracy • It may be expressed in terms of: • scale range.Limited value. • "Percentage of True Value". • Accuracy can be improved by calibration.

15. Limited error. • It is accuracy that is guaranteed by the manufacturer of some instrument or for some fabrication process. • Typically it can be expressed as some percent value of the full scale. • Example 1% of FSD. • It is constant for the instrument.

16. Limited error Example • 0 - 150 v voltmeter has a guaranteed accuracy of 1 percent of full scale reading. If the voltage measured by the instrument is 75 V. Calculate the Limiting error in percent.

17. Example…………. • But if we make a measurement of 10 V. • The relative error now is 15%.!!! • So we have to select the proper scale in order to obtain the maximum possible deflection. • Rule of thumb: more than 50% of FSD

18. Precision. • It is a measure of the reproducibility of the measurements. • It is a measure of the degree of agreement within a group of measurements that should be the same true value or that are making in the same conditions. • It is given by the maximal deviation respect to mean value of the measurement.

20. Accuracy and precision • If your instrument is accurate , has a good accuracy, it is also precise ( had a good precision). • But your instrument can be precise but the accuracy can be low, because some systematic error could be present Accuracy Precission

21. Static sensitivity It is given by the slope of the output/input curve

22. Linearity Y Maximum deviation “Best line” Least square fit X X X X X X X X

23. Linearity Example

24. LEAST SQUARE FIT SLOPE INTERCEPT LINEAR REGRESION COEFICIENT.

25. Hysteresis

26. Histéresis Y X

27. Threshold • If the instrument input is increased very gradually from zero, there will be some minimum value below which no output change can be detected. This minimum value defines the threshold of the instrument

28. Resolution • If the input is increased slowly from some arbitrary (nonzero) input value, again the output does not change at all until a certain input increment is exceeded

29. Zero drift

30. Sensitivity Drift

31. Zero & sensitivity drift

32. Random error processing. • We detect a random processing error, when we repeat measurements that are supposed to be constant ( they are realized in the same conditions) and we obtain a set of different values. X= { X1 X2 X3 X4 X5………….XN}

33. Random error Processing Result Standard deviation Mean value

34. Normal Probability Distribution Function PDF or Gaussian F • Depends on two parameters: mean value and standard deviation. • It is symmetric around to the mean value. • The integral of PDF in some interval determines the probability of values on this interval. • Probability is high around the mean value Only numerical calculation is possible.

35. Gaussian PDF

36. Total Error Calculation of Random Measurements • Random error and accuracy are statistical independent so this errors are squared added to get the total error. N We can reduce the random error as we want by increasing the number of measurement. But al last the total error is limited by the accuracy of your instrument

37. IMPORTANT MATLAB FUNCTIONS • <polytool> linear regression calculation. • <distool> PDF calculation. • <mean> mean value calculation • <std> standard deviation

38. Dynamic Characteristics • First order system. Described by first order differential equation. Impulse response Step response

39. First order System Step Response

40. Second order System It is described by a second order differential equation The response depends on the poles of the Transfer function.