Instrumentation (and Process Control) Uncertaintyee.bonabu.ac.ir/uploads/31/CMS/user/file/103/Instrumentation/Slides-Set-2-2.pdf•General rule: the process of measurement always disturbs

Instrumentation (and Process Control)

Fall 1393

Bonab University

Error

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Uncertainty

Measurement Uncertainty - Error

• 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

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Measurement Uncertainty - Error

• 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)

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Sources of Systematic Error

• 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 *

• Rm acts as a shunt

• Rm increase the ratio = 1

• Practical issues (increasing moving-coil instrument’s Rm)

• 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 circuit4

Error

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• 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:

RAB=500

measurement error = EO – Em = EO(1-9500/10000) = 0.05 EO 5%

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• Environmental 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

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• 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 mΩ/oC

• Care:

• Cross section (resistance)

• Route (not to pick up noise)

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Reduction of Systematic Errors

• 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

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• Method of Opposing Inputs• Example:

• If the coil resistance Rcoil 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 Rcomp

where Rcomp has a temperature coefficient equal in

magnitude but opposite in sign to that of the coil

• High-Gain Feedback• Unknown voltage Ei is applied to

• A motor of torque constant Km

• Resistance spring constant Ks

• Effect of environment on motor/spring = Dm/DS

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• High-Gain Feedback• No environment input: displacement Xo = KmKsEi , but changes with environment

• But if we close the loop:• Adding amplifier: Ka

• Feedback device: Kf

high Ka

• Only Kf ! we have to be concerned only with Df

• 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

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• 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)

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Quantification of Systematic Errors

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• 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


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• 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


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• Calculation of Overall Systematic Error• Total systemic error: often composed of several separate components

• measurement system loading

• Environmental factors

• Calibration errors

• worst-case prediction of maximum error: simply add up each separate systematic error• Example: 3 components of systematic error with a magnitude of ±1% each, a worst-case prediction

error: sum of the separate errors = ±3%

• However, it is very unlikely that all components be at their max/min simultaneously

• Usual course of action: combine separate sources root-sum-squares method

• n components:

• Warning: manufacturers data sheets measurement uncertainty/inaccuracy = best

estimate (manufacturer gives) about performance

• When it’s new, used under specified conditions, and recalibrated at the recommended

frequency


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• This can only be a starting point in estimating the measurement accuracy

(achievable in use)

• Many sources (systematic error) may apply in a particular situation (not

included in the accuracy calculation in the manufacturer’s data sheet)

• Example:• 3 separate sources of systematic error are identified in a measurement system

• After reducing the magnitude of these errors as much as possible, the magnitudes of the three errors

are estimated:

• System loading: +1.2% (Xm-Xt)

• Environmental changes: 0.8%

• Calibration error: 0.5%

• Calculate the maximum possible total systematic error and the likely system error by the root-sum-

square method.

• Solution:• The maximum possible system error = ±(1.2 + 0.8 + 0.5)% = ±2.5%

• Applying the root-sum-square: likely error = ± √1.22 + 0.82 + 0.52 = ±1.53%

Sources and Treatment of Random Errors

• Caused by unpredictable variations (precision errors)• Human observation

• electric noise

• random environmental changes (draught), etc.

• Small perturbations either side of the correct value (positive & negative errors occur in approximately equal numbers) largely eliminated by averaging • (but ≠ 0, reason: finite number of measurements)

• The degree of confidence (how close mean value is to the correct value)? • can be indicated by standard deviation or variance

• parameters describing distribution about the mean value

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Statistical Analysis of Measurements Subject to Random Errors

• Mean and Median Values• average value of a set of measurements of a constant quantity:

• Median (was easier for a computer to find) even number midway

• Mean (slightly closer to the correct value)

• Example:• length of a steel bar (mm) is measured by a number of different observers

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Set Mean Median

A 398 420 394 416 404 408 400 420 396 413 430 409 408

B 409 406 402 407 405 404 407 404 407 407 408 406 407

C 409 406 402 407 405 404 407 404 407 407 408 406.5 406

406 410 406 405 408 406 409 406 405 409 406 407

• Which one more

confidence?

• Low-Spread: say range

430-394=36 vs 409-402=7

• Median closer

to mean


• Standard Deviation and Variance• Spread = range between the largest and the smallest value not a very good way of

examining distribution

• Much better: variance or standard deviation start with deviation (error)

• di = xi – xmean

• Variance:

• Standard deviation:

• Definitions: infinite number of data not in practice

• Finite measurements: xmean ≠ true mean (µ)

• A better prediction: Bessel correction

• Finite number:

• Example:

• Previous sets of measurement

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Set Mean Median spread V σ

A 409 408 36 137 11.7

B 406 407 7 4.2 2.05

C 406.5 406 8 3.53 1.88

_ _


• Graphical Data Analysis Techniques—Frequency Distributions• Simplest: Histogram

• Bands/bins of equal width across the range of measurement

• # measurements within each band

• Finding the # of bands/bins (Sturgis Rule):

• Example: 23 measurements in set-C

• Bins: 5

• Span: 402-410mm

• Width? 2mm works

• Care in choice of boundaries:

• No measurements on the boundary

• Say, put the middle bin on the Mean (406.5)

• Large enough # of measurement & truly

Random error symmetry

• Usually, error is of most concern deviation19

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# Measurements

mm

Gaussian (Normal) Distribution

• Measurement sets that only contain random errors a distribution with a particular shape that is called Gaussian

• Frequency of small deviations from the mean >>

the frequency of large deviations

• Measurements in a data set subject to random

errors lie inside deviation boundaries of ±σ

68%

• Lie inside deviation boundaries of ±2σ

95.4%

• Lie inside deviation boundaries of ±3σ

99.7%

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±1.96σ 95% (Very common)

Error

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Standard Error of the Mean

• We examined: how measurements with random errors are distributed about the mean

• However, we know: error exists (mean value of a finite set - true value)

• The standard deviation of mean values of a series of finite

sets of measurements relative to the true mean = standard

error of the mean α

• Question:• if we use the mean value of a finite set of measurements to

predict the true value what is the likely error?

• S.d. of error = α 68% of deviations around true value within ±α

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Mean of 10

Standard Error of the Mean

• Question:• if we use the mean value of a finite set of measurements to

predict the true value what is the likely error?

• S.d. of error = α 68% of deviations around true value within ±α

• Means: with 68% certainty that the magnitude of the error

does not exceed |α|

• For data set C , n = 23, σ = 1.88 α = 0.39

• The length (average): 406.5 ± 0.4 (68% confidence limit)

• ±2α length : 406.5 ± 0.8 (95.4% confidence limit)

• ±3α length : 406.5 ± 1.2 (99.7% confidence limit)

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Mean of 10

Not so good

Estimation of Random Error in a Single Measurement (n=1)

• Usually, not practical (repeated measurements find average)• Or measured quantity is not constant

• What: likely magnitude of error?

• Often: calculate the error within 95%confidence limits ± 1.96σ

• However, it was only maximum likely deviation from calculated mean

• Not the true value

• Add: standard error of the mean to the likely maximum deviation value (95%)

• Example:• A standard mass is measured 30 times (same instrument), σ=0.46 α=0.08

• Now, measure an unknown mass 105.6 kg, how should the mass value be expressed?

• Solution:

• ± 1.96(σ+α) = ±1.06 mass: 105.6±1.06 kg

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Rogue Data Points (Data Outliers)

• Very large error measurements sometimes occur (random & unpredictable) • Error magnitude: much larger than the expected random variations

• Sources:• Sudden transient voltage surges

• Incorrect data recording

• Accepted practice:• Discard these data points

• Threshold: ±3σ

• Practical problem:• When a new dataset is measured (S.D. is not known)

• It’s possible to have outlier in measurements

• Simple solution:

• Any new set of measurement Histogram

• Examine to spot outliers

• Exclude if any calculate ±3σ to test future measurements24

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Boxplot

Error

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Uncertainty

Aggregation of Measurement System Errors

• Usually, 2 or more sources of measurement error

• Total likely error in output reading?

• Forms of aggregation:• A measurement have both: systematic (±x) & random errors (±y)

• Often likely maximum error:

• A measurement system have several measurement each with separate errors

• Say, different instruments/transducers add, subtract, multiply, divide

• Example: as S = y + z

• Problem: error term is not expressed as percentage of calculated value for S

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Aggregation of Measurement System Errors

• Problem: error term is not expressed as percentage of calculated value for S

• Statistical analysis:

• Example: A circuit requirement for a resistance of 550 (2 resistors of nominal values 220 and 330 in series)

• If each resistor has a tolerance of ±2%, the error in the sum?

• It can be shown that the error (e) for subtraction is the same

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Error

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Instrumentation (and Process Control) Uncertaintyee.bonabu.ac.ir/uploads/31/CMS/user/file/103/Instrumentation/Slides-Set-2-2.pdf•General rule: the process of measurement always disturbs

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