ELE 312 MEASUREMENTS AND INSTRUMENTATION by engr Dr
ELE 312 MEASUREMENTS AND INSTRUMENTATION by engr. Dr. (mrs. ) n. t. surajudeen-bakinde department of electrical and electronics engineering faculty of engineering and technology university of ilorin, ilorin LECTURE 3 THEORY OF ERRORS
OUTLINE �Introduction to Errors Classification of Errors o Sources of Errors o Reduction of Errors o Choosing appropriate measuring instruments o
Introduction to Errors An error is a deviation between the actual value of the measurand the indicated value proposed by the sensor or instrumentation used to measure the value. It is also the departure from the expected perfection. Ø Classification of Errors in measurement systems can be divided into those that arise during the measurement process and those that arise due to later corruption of the measurement signal by induced noise. Errors arising during the measurement process can be divided into two groups, known as systematic/Bias errors and Random errors.
q. Systematic or Bias errors �This is the difference between the average value in a series of repeated calibration measurements and the true value. �They are caused by imperfect calibration of measurement instruments or imperfect methods of observation, or interference of the environment with the measurement process.
q. Random errors This is a measure of the random variation found during repeated measurements. These errors are caused by the sudden change in experimental conditions(such as fluctuation in voltage, change in temperature or humidity), noise or tiredness of operators. �N. B: Random error corresponds to imprecision, and bias to inaccuracy
Sources of Errors �Sources of Systematic Errors Systematic errors in the output of many instruments are due to factors inherent in the manufacture of the instrument arising out of tolerances in the components of the instrument. There are two major sources of systematic errors which are: q. System disturbance due to measurement q. Errors due to environmental inputs Others include: Wear in instrument components and Observational Errors
Ø System disturbance due to measurement Disturbance of the measured system by the act of measurement is a common source of systematic error. If we were to measure the temperature of a beaker of hot water with a mercury-in-glass thermometer; by placing thermometer(which was a cold mass ), into the water, heat transfer occurs between the water and thermometer. This heat transfer would lower the temperature of the water, which causes a small reduction in temperature undetectable by the limited measurement resolution of such a thermometer. This clearly establishes the principle that, in nearly all measurement situations, the process of measurement disturbs the system and alters the values of the physical quantities being measured.
Ø Errors due to environmental inputs Those errors are caused by a change in the environmental conditions surrounding the measurement system. Environmental conditions such as pressure, temperature, humidity or even magnetic fields. ØWear in instrument components Systematic errors can frequently develop over a period of time because of wear in instrument components.
Ø Observational Errors These types of errors occurs due to wrong observations or reading in the instruments particularly in case of energy meter reading. These wrong observations may be due to parallax. Reduction of Errors The prerequisite for the reduction of systematic errors is a complete analysis of the measurement system that identifies all sources of error. The following are the various methods of reducing systematic errors: q Careful instrument design q Method of opposing inputs q High-gain feedback q Calibration q Manual correction of output reading q Intelligent instruments
Statistical Analysis of Measurements subject to Random Errors � q The following are the tools used in the Statistical analysis of measurement subject to random errors: Mean and Median Value: xmean = x 1+x 2+……xn n Xmedian = xn+1 2 Standard deviation and variance: d 1= x 1 – xmean q Variance (V) = d 12 +d 22+……d 2 n n-1 Standard deviation is the square root of variance. Standard deviation = √ d 12 +d 22+……d 2 n n-1
EXAMPLE 1 � Suppose that the components in the circuit shown in Figure 3. 1(a) have the following values: R 1=330; R 2=1000; R 3 =1200; R 4=220; R 5=270. If the instrument measuring the output voltage across AB has a resistance of 5000, what is the measurement error caused by the loading effect of this instrument?
Solution Internal resistance of the voltmeter = 5000 Using Thevenin’s theorem Total resistance is given as RAB = [(330+100)*1200 /(330+100+1200) +220]*270 [(330+100)*1200/(330+100+1200)] +220+270 RAB = 25. 597Ω
The voltage measured by the meter is then given by: In the absence of the measuring instrument and its resistance Rm, the voltage across AB would be the equivalent circuit voltage source whose value is E 0. The effect of measurement is therefore to reduce the voltage across AB by the ratio given by:
� The error is given by E 0 -Em: E 0 1 - Em =E 0 1 - 5000 25. 597+5000 Em = 0. 99 E 0 - E 0 Em =0. 01 or 1% - E 0 RM RAB+RM
EXAMPLE 2 � In a survey of 15 owners of a certain model of car, the following figures for average petrol consumption were reported. 25. 5 30. 3 31. 1 29. 6 32. 4 39. 4 28. 9 30. 0 33. 3 31. 4 29. 5 30. 5 31. 7 33. 0 29. 2 Calculate the mean value, the median value and the standard deviation of the data set.
Solution Mean=25. 5+30. 3+31. 1+29. 6+32. 4+39. 4+28. 9+30. 0+33. 3+31. 4+29. 5+30. 5+31. 7+33. 0+29. 2 15 xmean = 31. 05 Xmedian = xn+1 = 15+1 = 16 = 8 th number is the median number 2 2 Xmedian = 30. 0 Standard Deviation is the square root of variance. d 1= x 1 – xmean Variance (V) = d 12 +d 22+……d 2 n n-1 Ø Note n = 15
f(x) Xmean 25. 5 31. 05 -5. 55 30. 3 31. 05 -0. 75 31. 1 31. 05 0. 05 29. 6 31. 05 -1. 45 32. 4 31. 05 1. 35 39. 4 31. 05 8. 35 . . 29. 2 Total(d 12+d 22+…… d 2 n ) Variance (V) = (126. 24)2 = 9. 017143 14 Deviation(dx) 31. 05 -1. 85 () 126. 24
Standard Deviation =√ d 12 +d 22+……d 2 n = √ 9. 017143 n-1 Standard Deviation = 1. 276
Practise Questions �Practise all questions at the end of Chapter 3 of the text book. Measurement and Instrumentation Principles by Alan S. Morris, Third Edition, Published by Butterworth Heineman , 2001.
- Slides: 19