Instrumentation (and Error & Process Control) Uncertainty Fall 1393 Bonab University
Error Measurement Uncertainty - Error & Uncertainty • Measurement errors are impossible to avoid • We can minimize their magnitude by • Good measurement system design • Appropriate analysis and processing of measurement data All error sources: How to eliminate or reduce their magnitude • • Errors: • Arise during the measurement process * • Arise due to later corruption of the measurement signal (by induced noise during transfer of the signal) • In any measurement system it’s important to: • Reduce errors to the minimum possible level • quantify the maximum remaining error that may exist in output reading • What if system final output is calculated by combining together two or more measurements? • How each separate measurement error be combined best estimate of the final output error • Error main categories: • Systematic • Random 2
Error Measurement Uncertainty - Error & Uncertainty • Systematic: • Describe errors in the output readings that are consistently on one side of the correct reading, that is, either all errors are positive or are all negative • System disturbance during measurement • The effect of environmental changes • Bent needles, use of uncalibrated instruments, drift, poor cabling, … • The remaining is quantified by the quoted accuracy • Random: • (precision errors) are perturbations of the measurement in either side of the true value caused by random and unpredictable effects, such that positive errors and negative errors occur in approximately equal numbers • mainly small, but large perturbations occur from time to time • Human observation of analog device + interpolation • Electrical noise • Can be largely removed by: many measurements averaging or other statistical techniques • The best way is to express them in probabilistic terms (say, 95% CI) 3
Error Sources of Systematic Error & Uncertainty • Disturbance of the measured system by the act of measurement • Mercury-in-glass thermometer • Orifice plate • General rule: the process of measurement always disturbs system being measured • Accurate understanding of the mechanisms of system disturbance is needed to minimize it • Case: Electric circuits: • The´ venin’s theorem * • R m acts as a shunt • R m increase the ratio = 1 • Practical issues (increasing moving- coil instrument’s R m ) • Solve: changing the spring constant • Ruggedness changes, and needs better friction • So, usually improving one aspect introduce another problem • Using active devices improves this limit • Case: measuring instrument in a bridge circuit 4
Error Sources of Systematic Error & Uncertainty • Example: R1 = 400 O; R2 = 600 O; R3 = 1000 O; R4 = 500 O; R5 = 1000 O The voltage across AB is measured by a voltmeter whose internal resistance is 9500 O. What is the measurement error caused by the resistance of the measuring instrument? • Solution: R AB =500 measurement error = E O – E m = E O (1-9500/10000) = 0.05 E O 5% 5
Error Sources of Systematic Error & Uncertainty • E nvironmental disturbances (modifying inputs) • static and dynamic characteristics specified for measuring instruments are only valid for particular environmental conditions • These specified conditions must be reproduced during calibration • Its magnitude quantified by: • sensitivity drift • zero drift (both included in the specifications) • Env. Disturbance is difficult to determine • Example: A small closed box (0.1 kg) scale says 1kg (a) a 0.9 kg rat in the box (real input) (b) an empty box with a 0.9 kg bias on the scale due to a temperature change (environmental input) (c) a 0.4 kg mouse in the box together with a 0.5 kg bias (real þ environmental inputs) • Thus, the magnitude of any environmental input must be measured before the value of the measured quantity (the real input) can be determined from the output reading of an instrument • Designers’ choice: • Reduce the susceptibility of measuring instruments to environmental inputs • Quantify the effects of environmental inputs and correct for them in the instrument output reading 6
Error Sources of Systematic Error & Uncertainty • Changes in characteristics due to wear in instrument components (with time) • Systematic errors can frequently develop over a period of time because of wear in instrument components • Recalibration often provides a full solution • Resistance of connecting leads • Example: a resistance thermometer • Often thermometer is separated by 100 meters (20-gauge copper wire is 7 Ω ) • Also: a temperature coefficient of 1 m Ω / o C • Care: • Cross section (resistance) • Route (not to pick up noise) 7
Error Reduction of Systematic Errors & Uncertainty • Prerequisite : a complete analysis of the measurement system that identifies all sources of error • Simple faults: bent meter needles, poor cabling practices… • other error sources require more detailed analysis and treatment • Careful Instrument Design • Reducing the sensitivity (strain gauge to temperature) cost • Calibration • All instruments suffer from drift in their characteristics it depends on: • environmental conditions • Frequency of use • More frequent calibration = lower drift-related error • Method of Opposing Inputs • compensates : effect of an environmental input by introducing an equal and opposite environmental input that cancels it out 8
Error Reduction of Systematic Errors & Uncertainty • Method of Opposing Inputs • Example: • If the coil resistance R coil is sensitive to temperature, environmental input ( temperature change) will alter the value of the coil current for a given applied voltage alter the pointer output reading • Compensation: introducing a compensating resistance R comp where Rcomp has a temperature coefficient equal in magnitude but opposite in sign to that of the coil • High-Gain Feedback • Unknown voltage E i is applied to • A motor of torque constant K m • Resistance spring constant K s • Effect of environment on motor/spring = D m /D S 9
Error Reduction of Systematic Errors & Uncertainty • High-Gain Feedback • No environment input: displacement X o = K m K s E i , but changes with environment • But if we close the loop: • Adding amplifier: K a • Feedback device: K f high K a • Only K f ! we have to be concerned only with D f • Signal Filtering • corruption of reading by periodic noise • often at a frequency of 50 Hz caused by pickup through the close proximity to apparatus or current-carrying cables • High frequency noise (mechanical oscillation/vibration) • Appropriate filter (LP , BP , BS) reduces noise amplitude 10
Error Reduction of Systematic Errors & Uncertainty • Signal Filtering • Example: passive RC LP filter • Manual Correction of Output Reading • Errors due to • system disturbance during the act of measurement • Environmental changes • a good measurement technician reduce errors • by calculating the effect of such systematic errors • making appropriate correction to readings • Not easy (needs all disturbances quantified) • Intelligent Instruments • Contain extra sensors that measure the value of environmental inputs • Automatically compensate the value of the output reading • ability to deal very effectively with systematic errors (explained later) 11
Error Quantification of Systematic Errors & Uncertainty • Do all practical steps have been taken to eliminate or reduce the magnitude of systematic errors? quantify the maximum likely systematic error • Quantification of Individual Systematic Error Components • first complication: exact value for a component = ? use best estimate: • Environmental condition errors • Environment effect? Assume midpoint environmental conditions • • specify maximum measurement error as ±x% of the output reading If fluctuations occur over a short period of time (random draughts of hot or cold air) this is a • rather a random error • Calibration errors • The maximum error just before the instrument is due for recalibration becomes the basis for estimating the maximum likely error 12
Error Quantification of Systematic Errors & Uncertainty • Calibration errors • Example (a pressure transducer ): • Recalibration frequency: once the measurement error has grown to +1% of the full-scale Range: 0 to 10 bar • • How can its inaccuracy be expressed in the form of a ±x% error in the output reading? • Solution: • Just before recalibration: error grown to +0.1 bar (1% of 10 bar) • Half this maximum error, 0.05 bar, should be subtracted from all measurements • Error: • just after calibration: -0.05 bar ( -0.5% of FSR) • just before the next recalibration: +0.05 bar (+0.5% of FSR) • Inaccuracy due to calibration error: ±0.05% of FSR • System disturbance (as well as loading) errors • Maximum likely error = 2x (worst-case system loading) Likely error: ±x ±y% FSD 13
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