Engineering Measurements Measurement Uncertainty - Part 01 PDF Fall 2024

Systematic Errors Random Errors ENGN-3220 Engineering Measurements Measurement Uncertainty - Part 01 Fall 2024 ENGN-3220: Engineering Measurements 1/36  Systematic Errors Random Err...

Systematic Errors Random Errors ENGN-3220 Engineering Measurements Measurement Uncertainty - Part 01 Fall 2024 ENGN-3220: Engineering Measurements 1/36  Systematic Errors Random Errors Introduction Measurement errors are impossible to avoid, although we can minimize their magnitude by good design of the measurement system accompanied by appropriate analysis and processing of measurement data. Errors arising during the measurement process can be divided into two groups, known as systematic errors and random errors. ENGN-3220: Engineering Measurements 2/36  Systematic Errors Random Errors Introduction Systematic errors describe errors in the output readings of a measurement system that are consistently on one side of the correct reading, that is, either all errors are positive or are all negative. Two major sources of systematic errors are system disturbance during measurement and the effect of environmental changes. ENGN-3220: Engineering Measurements 3/36  Systematic Errors Random Errors Introduction Other sources of systematic error include use of uncalibrated instruments, drift in instrument characteristics, and poor cabling practices. Even when systematic errors due to these factors have been reduced or eliminated, some errors remain which are inherent in the manufacturing of an instrument; such errors are quantified by the accuracy value. ENGN-3220: Engineering Measurements 4/36  Systematic Errors Random Errors Introduction Random errors, which are also called precision errors, are perturbations of the measurement either side of the true value caused by random and unpredictable effects, such that positive errors and negative errors occur in approximately equal numbers for a series of measurements made of the same quantity. Such perturbations are mainly small, but large perturbations occur from time to time, again unpredictably. Random errors often arise due to human errors and also electrical noise. ENGN-3220: Engineering Measurements 5/36  Systematic Errors Random Errors Introduction To a large extent, random errors can be overcome by taking the same measurement a number of times and extracting a value by averaging or other statistical techniques. Wherever a systematic error exists alongside random errors, correction has to be made for the systematic error in the measurements first before statistical techniques are applied to reduce the effect of random errors. ENGN-3220: Engineering Measurements 6/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error Systematic Errors The main sources of systematic error in the output of measuring instruments can be summarized as: System disturbance due to the measurement process itself Effect of environmental disturbances (“sensitivity drift” and “zero drift”) Changes in instrument characteristics due to mechanical wear of the instrument components over a period of time Resistance of connecting leads Systematic errors can be reduced by careful instrument design, calibration, signal filtering (to remove noise), and the use of intelligent instruments. ENGN-3220: Engineering Measurements 7/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement Disturbance of the measured system by the act of measurement is a common source of systematic error. In nearly all measurement situations, the process of measurement itself disturbs the system (which is being measured) and alters the values of the physical quantities being measured. Example: measuring the temperature of beaker of hot water... The magnitude of the disturbance varies from one measurement system to the other and is affected particularly by the type of instrument used for measurement. ENGN-3220: Engineering Measurements 8/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement An accurate understanding of the mechanisms of system disturbance is needed in order to properly design the measuring instrument in such a way to minimize the disturbance of the system being measured. ENGN-3220: Engineering Measurements 9/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement For instance, consider the circuit shown below in which the voltage across resistor R5 is to be measured by a voltmeter with internal resistance Rm. ENGN-3220: Engineering Measurements 10/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement Voltmeters are connected in parallel with the component across which you want to measure the voltage. Attaching the voltmeter to the circuit changes the circuits resistance because Rm gets in parallel with R5 hence affecting the voltage E0 which is to be measured. Thevenin’s theorem can help us in analyzing this situation in order to minimize the disturbance. ENGN-3220: Engineering Measurements 11/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement ENGN-3220: Engineering Measurements 12/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement It can be shown easily that the Thevenin equivalent resistance RAB is given as RAB  prpR1 R2 q||R3 s R4 q ||R5 Hence the measured voltage Em can be expressed in terms of the original (un-disturbed) voltage E0 as follows Em  E0 R Rm RAB  E0 1 RAB m 1 Rm from which we can conclude that as Rm increases, the measured voltage Em gets closer to the original voltage E0. ENGN-3220: Engineering Measurements 13/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement Therefore the design strategy of a voltmeter should be to make its internal resistance Rm as high as possible to minimize disturbance of the system being measured. ENGN-3220: Engineering Measurements 14/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement ENGN-3220: Engineering Measurements 15/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement The Thevenin’s equivalent resistance can be computed as follows RAB  prpR1 R2 q||R3 s R4 q ||R5  0.5 kΩ The measured voltage Em can be expressed in terms of the un-disturbed voltage E0 as follows Em  E0 R Rm RAB  E0 9.5 kΩ 10 kΩ  0.95E0 m Hence the measurement error is E0
Em 100  E0  100  E0
E0.95E0  5% 0 ENGN-3220: Engineering Measurements 16/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement As an another example, let’s consider the circuit shown below in which the current flowing in the resistor R5 is to be measured by an ammeter with internal resistance Rm. ENGN-3220: Engineering Measurements 17/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement Adding the ammeter (in series with R5 ) will make the circuit looks as shown below. ENGN-3220: Engineering Measurements 18/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement Attaching the ammeter to the circuit changes the circuits resistance because Rm gets in series with R5 hence affecting the current I0 which is to be measured. Thevenin’s theorem can help us in analyzing this situation in order to minimize the disturbance. ENGN-3220: Engineering Measurements 19/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement ENGN-3220: Engineering Measurements 20/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement It can be shown easily that the Thevenin equivalent resistance RT h is given as RT h  rpR1 R2 q||R3 s R4 R5 The un-disturbed current I0 may be computed as follows: I0  REs Th ENGN-3220: Engineering Measurements 21/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error System Disturbance due to Measurement The measured current Im can be expressed as follows Im R Es Rm  I0 1 Rm Th 1 RT h from which we can conclude that as Rm decreases, the measured current Im gets closer to the original current I0. Therefore the design strategy of a ammeter should be to make its internal resistance Rm as small as possible to minimize disturbance of the system being measured. ENGN-3220: Engineering Measurements 22/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error Calculation of Overall Systematic Error The total systemic error in a measurement is often composed of several separate components, for example, measurement system loading, environmental factors, and calibration errors. A worst-case prediction of maximum error would be to simply add up each separate systematic error. For example, if there are three components of systematic error with a magnitude of 1% each, a worst-case prediction would be the sum of the separate errors, that is 3%. ENGN-3220: Engineering Measurements 23/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error Calculation of Overall Systematic Error However, it is very unlikely that all components of error would be at their maximum or minimum values simultaneously. The usual course of action is therefore to combine separate sources of systematic error using a root-sum-squares method. Applying this method for n systematic component errors of magnitude x1 %, x2 %,..., xn %, the best prediction of likely maximum systematic error by the root-sum-squared method is error   a|x | 2 |x2|2... |xn|2 1 ENGN-3220: Engineering Measurements 24/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error Calculation of Overall Systematic Error ENGN-3220: Engineering Measurements 25/36  Systematic Errors System Disturbance due to Measurement Random Errors Calculation of Overall Systematic Error Calculation of Overall Systematic Error ENGN-3220: Engineering Measurements 26/36  Systematic Errors Random Errors Random Errors Random errors in measurements are caused by unpredictable variations in the measurement system. Random errors are some times referred to as precision errors. Typical sources of random error are: Measurements taken by human observation of an analogue meter, especially where this involves interpolation between scale points. Electrical noise. Random environmental changes. ENGN-3220: Engineering Measurements 27/36  Systematic Errors Random Errors Random Errors Random errors are usually observed as small perturbations of the measurement either side of the correct value, that is, positive errors and negative errors which occur in approximately equal numbers for a series of measurements made of the same constant quantity. Therefore, random errors can largely be eliminated by calculating the average of a number of repeated measurements of the same quantity (which is assumed to be constant during the repeated measurements). This averaging process of repeated measurements can be done automatically by intelligent instruments. ENGN-3220: Engineering Measurements 28/36  Systematic Errors Random Errors Random Errors The mean value of a set of n measurements is defined as ņ xmean  1 n i1 xi A quantity called the “standard deviation” of a set of n measurements xi is defined as g f σx f  e n1 ņ pxi
xmeanq2  i 1 ENGN-3220: Engineering Measurements 29/36  Systematic Errors Random Errors Random Errors The standard deviation σx represents the dispersion or deviation of the data points (i.e. measured quantities) from the mean value. A related quantity is the “variance” of the set of n measurements xi , the variance is simply equal to σx2 , that is ņ σx2  n1 pxi
xmeanq2 i 1  ENGN-3220: Engineering Measurements 30/36  Systematic Errors Random Errors Random Errors Those formal definitions for the variance and standard deviation of data are made with respect to an infinite population of data values, whereas in all practical situations, we can only have a finite set of measurements. A better prediction of the variance can be obtained by applying a correction factor known as the Bessel correction factor: g f σx f  e n
1 1 ņ pxi
xmeanq2  i 1 ENGN-3220: Engineering Measurements 31/36  Systematic Errors Random Errors Random Errors Example Calculate the standard deviation for the following data sets A, B and C ENGN-3220: Engineering Measurements 32/36  Systematic Errors Random Errors Random Errors ENGN-3220: Engineering Measurements 33/36  Systematic Errors Random Errors Random Errors ENGN-3220: Engineering Measurements 34/36  Systematic Errors Random Errors Random Errors ENGN-3220: Engineering Measurements 35/36  Systematic Errors Random Errors References 1 Morris, Alan S., and Reza Langari. Measurement and instrumentation: theory and application. Academic Press, 2021. 2 Morris, Alan S., “Measurement and Instrumentation Principles”, 3rd Edition, Butterworth-Heinemann, 2001 ENGN-3220: Engineering Measurements 36/36