electrical & electronics measurement ee302 · 2012-06-27 · loading effects ii.. underunder...

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ELECTRICAL & ELECTRONICS MEASUREMENT EE302

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An OverviewAn Overview

Measurements For Nation’s Progress Is Imperative: Measurements For Nation’s Progress Is Imperative: Why?Why?

TheThe advancementadvancement ofof ScienceScience andand TechnologyTechnology isis dependentdependent uponupon

aa parallelparallel progressprogress inin measurementsmeasurements techniquestechniques.. TheThe reasonreason isis

thatthat thethe progressprogress inin sciencescience andand technologytechnology leadsleads toto discoveringdiscovering

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newnew phenomenaphenomena andand relationships,relationships, whichwhich requiresrequires newnew typestypes ofof

quantitativequantitative measurementsmeasurements..

Field Of Field Of Engineering ApplicationsEngineering Applications Of Measurement SystemsOf Measurement Systems

1.1. DesignDesign ofof equipmentequipment andand processesprocesses..

2.2. ProperProper operationoperation andand maintenancemaintenance ofof equipmentequipment andand processesprocesses..

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3.3. QualityQuality controlcontrol assuranceassurance programsprograms forfor industrialindustrial processesprocesses..

Methods of MeasurementsMethods of Measurements

1.1. DirectDirect MethodsMethods:: TheThe unknownunknown quantityquantity isis directlydirectly comparedcompared againstagainst

aa standard,standard, whichwhich areare commoncommon forfor thethe measurementsmeasurements ofof physicalphysical

quantitiesquantities likelike length,length, massmass andand timetime..

DisadvantagesDisadvantages areare::

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DisadvantagesDisadvantages areare::

i.i. LimitedLimited accuracyaccuracy duedue toto humanhuman factorsfactors..

ii.ii. LessLess sensitivesensitive..

Methods of MeasurementsMethods of Measurements, , contcont

2.2. IndirectIndirect MethodsMethods:: TheThe unknownunknown quantityquantity underunder measurementmeasurement isis

determineddetermined viavia thethe useuse ofof measurementmeasurement systemssystems asas followsfollows::

AdvantagesAdvantages areare::

to be to be measuredmeasured

QuantityQuantity

TransducerTransducer

For converting For converting

physical quantity physical quantity

into electric into electric

signalsignal

Signal Signal

ProcessingProcessing

For noise For noise

reduction, reduction,

amplification, etcamplification, etc

Measuring Measuring

DeviceDevice

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AdvantagesAdvantages areare::

i.i. HighHigh accuracyaccuracy andand sensitivitysensitivity cancan bebe obtainedobtained byby usingusing electronicelectronic

andand digitaldigital typetype instrumentsinstruments..

ii.ii. AvailabilityAvailability toto useuse commercialcommercial instrumentinstrument--typestypes atat lowerlower costscosts withwith

acceptedaccepted accuracyaccuracy andand sensitivitysensitivity..

iii.iii. MeasurementsMeasurements ofof nonnon--physicalphysical quantitiesquantities..

Classification of InstrumentsClassification of Instruments

Broadly, instruments are classified into two categories:Broadly, instruments are classified into two categories:

1.1. AbsoluteAbsolute instrumentsinstruments:: TheseThese instrumentsinstruments givegive thethe magnitudemagnitude ofof thethe

quantityquantity toto bebe measuredmeasured inin termsterms ofof physicalphysical constantsconstants ofof thethe

instrumentsinstruments.. ForFor example,example, TangentTangent GalvanometerGalvanometer.. ItIt isis usedused onlyonly inin

standardstandard institutionsinstitutions forfor calibrationcalibration..

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standardstandard institutionsinstitutions forfor calibrationcalibration..

2.2. SecondarySecondary instrumentsinstruments:: InIn thesethese instrumentsinstruments thethe magnitudemagnitude ofof thethe

quantityquantity toto bebe measuredmeasured isis indicatedindicated onon gradedgraded scalescale (e(e..gg.. analoganalog

instruments)instruments) oror displayeddisplayed numericallynumerically onon screenscreen (e(e..gg.. digitaldigital

instruments)instruments)..

Errors in MeasurementsErrors in Measurements

1.1. True Value:True Value:

i.i. ItIt isis notnot possiblepossible toto determinedetermine thethe “true“true value”value” ofof aa quantityquantity byby

experimentalexperimental meansmeans.. TheThe reasonreason forfor thisthis isis thatthat thethe positivepositive

deviationsdeviations fromfrom thethe truetrue valuevalue dodo notnot equalequal thethe negativenegative deviationsdeviations

andand hencehence dodo notnot cancelcancel eacheach otherother..

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ii.ii. InIn practicepractice thethe “true“true value”value” isis measuredmeasured byby aa ““standardstandard unitunit””.

2.2. Static (or Absolute) ErrorStatic (or Absolute) Error ( )Aδ

ItIt isis defineddefined asas thethe deviationdeviation ofof thethe measuredmeasured valuevalue (A(Amm)) fromfrom itsits truetrue

oneone (A(Att),), oror tm AAA −=δ


3.3. Static CorrectionStatic Correction ( )Cδ

i.i. ItIt isis defineddefined asas thethe correctioncorrection toto bebe addedadded toto thethe measuredmeasured valuevalue soso

asas toto obtainobtain itsits truetrue value,value, oror

ii.ii. TheThe truetrue valuevalue ofof anan instrumentinstrument

mt AAC −=δ

CAA mt δ+=

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ii.ii. TheThe truetrue valuevalue ofof anan instrumentinstrument

iii.iii. Generally,Generally,

CAA mt δ+=

AC δ−=δ


4.4. Relative Static ErrorRelative Static Error ( )Aδ

i.i. TheThe percentagepercentage staticstatic errorerror ((%% εεεεεεεεrr)) ofof anan instrumentinstrument ==

ii.ii. WhenWhen AAtt isis unknown,unknown, thethe percentagepercentage staticstatic errorerror ((%%εεεεεεεεrr)) ofof anan

instrumentinstrument cancan bebe expressedexpressed asas aa fractionfraction ofof thethe fullfull scalescale deflectiondeflection

100A

A

t

×δ

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instrumentinstrument cancan bebe expressedexpressed asas aa fractionfraction ofof thethe fullfull scalescale deflectiondeflection

((ff..ss..dd)) asas::

iii.iii. TheThe truetrue valuevalue AAtt ==r

m

1

A

ε+

100d.s.f

A% r ×

δ=ε

Example Example 11::

AA voltagevoltage hashas aa truetrue valuevalue ofof 11..55 VV.. AnAn analoganalog indicatingindicating instrumentinstrument withwith

aa scalescale rangerange ofof 00--22..55 VV showsshows aa voltagevoltage ofof 11..4646 VV.. WhatWhat areare thethe valuesvalues

ofof absoluteabsolute errorerror andand correctioncorrection.. ExpressExpress thethe errorerror asas aa fractionfraction ofof thethe

truetrue valuevalue andand thethe ff..ss..dd..

Solution Solution 11::

V04.05.146.1AAAerrorAbsolute −−−−====−−−−====−−−−====δδδδ

11

%6.11005.2

04.0)d.s.fofpercentageaasressed(experrorlativeRe

%66.21005.1

04.0

A

A)(%errorlativeRe%

V04.0ACcorrectionAbsolute

V04.05.146.1AAAerrorAbsolute

tr

tm

−−−−====××××−−−−

====

−−−−====××××−−−−

====δδδδ

====εεεε

++++====δδδδ−−−−====δδδδ

−−−−====−−−−====−−−−====δδδδ


5.5. Limiting ErrorLimiting Error

TheThe manufacturersmanufacturers havehave toto specifyspecify thethe deviationdeviation fromfrom thethe specifiedspecified valuevalue

ofof aa particularparticular quantityquantity inin orderorder toto enableenable thethe purchaserpurchaser toto makemake properproper

selectionselection accordingaccording toto hishis requirementsrequirements.. TheThe limitslimits ofof thesethese deviationsdeviations

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fromfrom thethe specifiedspecified valuevalue areare defineddefined asas limitinglimiting errorserrors..


TheThe inductanceinductance ofof anan inductorinductor isis specifiedspecified byby aa manufacturermanufacturer asas 2020 HH ±±±±±±±± 55%%..

DetermineDetermine thethe limitslimits ofof inductanceinductance betweenbetween whichwhich itit isis guaranteedguaranteed..


05.05

)(errorlativeRe r ========εεεε

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(((( ))))(((( )))) Henery12005.0120

1AAA

AA)A(cetaninducofvalueLimiting

05.0100

)(errorlativeRe

rmmrm

m

r

±±±±====±±±±====

εεεε±±±±====εεεε±±±±====

δδδδ±±±±====

========εεεε


AA 00--2525 AA ammeterammeter hashas aa guaranteedguaranteed accuracyaccuracy ofof 11 percentpercent ofof fullfull

scalescale readingreading.. TheThe currentcurrent measuredmeasured byby thisthis instrumentinstrument isis 1010AA..

DetermineDetermine thethe limitinglimiting errorerror inin percentagepercentage..


amperes25.02501.0d.s.fA

,instrumenttheoferroritinglimofmagnitudeThe

r =×=×ε=δ

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025.010

25.0

A

A)valuemeasuredof(

amperes25.02501.0d.s.fA

m

r

r

==δ

=ε

=×=×ε=δ

Therefore, the current being measured is between the limits of

(((( )))) (((( ))))

%5.210010

25.0errorLimiting%

amperes25.010025.01101AA rm

====××××====∴∴∴∴

±±±±====±±±±====εεεε±±±±====

Errors in Measurements: Errors in Measurements: Random errorsRandom errors

i.i. TheseThese errorserrors areare ofof variablevariable magnitudemagnitude andand signsign andand dodo notnot obeyobey anyany

rulerule..

ii.ii. TheThe presencepresence ofof randomrandom errorserrors becomesbecomes evidentevident whenwhen differentdifferent resultsresults

areare obtainedobtained onon repeatedrepeated measurementsmeasurements ofof oneone andand thethe samesame quantityquantity..

iii.iii. TheThe effecteffect ofof randomrandom errorserrors isis minimizedminimized byby measuringmeasuring thethe givengiven

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quantityquantity manymany timestimes underunder thethe samesame conditionsconditions andand calculatingcalculating thethe

arithmeticarithmetic meanmean ofof thethe valuesvalues obtainedobtained..

iv.iv. TheThe problemproblem ofof randomrandom errorserrors isis treatedtreated mathematicallymathematically asas oneone ofof thethe

probabilityprobability andand statisticsstatistics..

be the number of nbe the number of n--repeated measurements, repeated measurements,

then the arithmetic mean equalsthen the arithmetic mean equals

Standard deviation is Standard deviation is

Errors in Measurements: Errors in Measurements: Random errorsRandom errors

n21 x,,x,xLet L

n

x

x

n

1i

i∑∑∑∑========

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Probable error =Probable error =

1n

xn

1i

2i

−=σ∑=

n

6745.0 σσσσ××××

Static Characteristics of InstrumentsStatic Characteristics of Instruments

1.1. AccuracyAccuracy

ItIt isis thethe closenesscloseness withwith whichwhich anan instrumentinstrument readingreading approachesapproaches thethe

truetrue valuevalue ofof thethe quantityquantity beingbeing measuredmeasured..

AccuracyAccuracy (in(in percent)percent) == r%100 εεεε−−−−

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r


2.2. SensitivitySensitivity

i.i. TheThe staticstatic sensitivitysensitivity ofof anan instrumentinstrument isis thethe ratioratio ofof thethe magnitudemagnitude

ofof thethe outputoutput signalsignal toto thethe magnitudemagnitude ofof thethe inputinput signalsignal oror thethe

quantityquantity toto bebe measuredmeasured..

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ii.ii. ItsIts unitsunits areare millimetermillimeter perper micromicro--ampereampere,, countscounts perper voltvolt,, etcetc..

dependingdepending uponupon thethe typetype ofof inputinput andand outputoutput signalssignals..

iii.iii. DeflectionDeflection factorfactor (or(or InverseInverse sensitivity)sensitivity) ofof anan instrumentinstrument isis thethe

reciprocalreciprocal ofof thethe sensitivitysensitivity ofof thatthat instrumentinstrument..


AA WheatstoneWheatstone bridgebridge requiresrequires aa changechange ofof 77 ΩΩΩΩΩΩΩΩ inin thethe unknownunknown armarm

ofof thethe bridgebridge toto produceproduce aa changechange inin deflectiondeflection ofof 33 mmmm ofof thethe

galvanometergalvanometer.. DetermineDetermine thethe sensitivitysensitivity.. AlsoAlso determinedetermine thethe

deflectiondeflection factorfactor..


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mm/33.2429.0

1

ySensitivit

1factorDeflection

/mm429.07

mm3ySensitivit

ΩΩΩΩ============

ΩΩΩΩ====ΩΩΩΩ

====

3.3. ResolutionResolution

ItIt isis thethe smallestsmallest incrementincrement inin inputinput (quantity(quantity beingbeing measured)measured) whichwhich cancan bebe

detecteddetected withwith certaintycertainty byby anan instrumentinstrument..



AA movingmoving coilcoil voltmetervoltmeter hashas aa uniformuniform scalescale withwith 100100 divisions,divisions, thethe fullfull--scalescale

readingreading isis 200200 VV andand 11//1010 ofof aa scalescale divisiondivision cancan bebe estimatedestimated withwith aa fairfair

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readingreading isis 200200 VV andand 11//1010 ofof aa scalescale divisiondivision cancan bebe estimatedestimated withwith aa fairfair

degreedegree ofof certaintycertainty.. DetermineDetermine thethe resolutionresolution ofof thethe instrumentinstrument inin voltsvolts..


V2.0210

1divisionscale

10

1solutionRe

V2100

200divisionscale1

====××××========

========


AA digitaldigital voltmetervoltmeter hashas aa readread--outout rangerange fromfrom 00 toto 99999999 countscounts..

DetermineDetermine thethe resolutionresolution ofof thethe instrumentinstrument inin voltsvolts whenwhen thethe fullfull

scalescale readingreading isis 99..999999 VV..


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mV1V10volt999.99999

1count

9999

1solutionRe

9999incount1isinstrumentthisofresolutionThe

3 ==×== −

Accuracy Accuracy versusversus PrecisionPrecision

i.i. TheThe termterm “Precise”“Precise” meansmeans clearlyclearly oror sharplysharply defineddefined..

ii.ii. PrecisionPrecision isis aa measuremeasure ofof thethe reproducibilityreproducibility (( oror consistency)consistency) ofof thethe

measurements,measurements, ii..ee.. givengiven aa fixedfixed valuevalue ofof aa quantity,quantity, precisionprecision isis aa

measuremeasure ofof thethe degreedegree ofof agreementagreement withinwithin aa groupgroup ofof measurementsmeasurements..

iii.iii. ConsiderConsider thethe measurementmeasurement ofof aa knownknown voltagevoltage ofof 100100 VV withwith aa metermeter..

FiveFive readingsreadings areare taken,taken, andand thethe indicatedindicated valuesvalues areare 104104,, 103103,, 105105,, 103103

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andand 105105 VV.. FromFrom thesethese valuesvalues itit isis seenseen thatthat

TheThe instrumentinstrument cannotcannot bebe dependeddepended onon forfor anan accuracyaccuracy betterbetter thanthan 55%%,,

WhileWhile aa precisionprecision ofof ±±±±±±±±11%% isis indicatedindicated sincesince thethe maximummaximum deviationdeviation

fromfrom thethe meanmean readingreading ofof 104104 VV isis onlyonly 11 VV..

%5100V100

V5errorLimiting% ====××××

++++====

WhenWhen aa numbernumber ofof independentindependent measurementsmeasurements areare takentaken inin orderorder toto

obtainobtain thethe bestbest measuredmeasured valuevalue,, thethe resultresult isis usuallyusually expressedexpressed asas

arithmeticarithmetic meanmean ofof allall readingsreadings.. TheThe rangerange ofof doubtdoubt oror possiblepossible errorerror isis

thethe largestlargest deviationdeviation fromfrom thethe meanmean ..

Range of Possible ErrorsRange of Possible Errors

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AA setset ofof independentindependent currentcurrent measurementsmeasurements werewere recordedrecorded asas 1010..0303,,

1010..1010,, 1010..1111 andand 1010..0808 AA.. CalculateCalculate thethe rangerange ofof possiblepossible errorserrors..


A11.10IcurrentofvalueMaximum

A08.104

08.1011.1010.1003.10

4

IIIIIcurrentAverage

max

4321av

====

====++++++++++++

====++++++++++++

====

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A04.02

05.003.0errorsofrangeAverage

A05.0IIRange

A03.10IcurrentofvalueMinimum

A03.0IIRange

A11.10IcurrentofvalueMaximum

minav

min

avmax

max

±±±±====++++

====∴∴∴∴

====−−−−====

====

====−−−−====

====

Loading EffectsLoading Effects

i.i. UnderUnder practicalpractical conditionsconditions thethe introductionintroduction ofof anyany measuringmeasuring

instrumentinstrument inin aa systemsystem results,results, invariably,invariably, inin extractionextraction ofof energyenergy

fromfrom thethe systemsystem therebythereby distortingdistorting thethe originaloriginal signalsignal underunder

measurementmeasurement..

ii.ii. ThisThis distortiondistortion maymay taketake thethe formform ofof attenuationattenuation (reduction(reduction inin

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magnitude),magnitude), waveformwaveform distortion,distortion, phasephase shiftshift andand manymany aa timetime allall

thesethese undesirableundesirable featuresfeatures putput togethertogether..

iii.iii. TheThe incapabilityincapability ofof thethe systemsystem toto faithfullyfaithfully measure,measure, record,record, oror

controlcontrol thethe inputinput signalsignal ((measurandmeasurand)) inin undistortedundistorted formform isis calledcalled

loadingloading effecteffect..


1.1. Loading Effects due to Loading Effects due to Shunt ConnectedShunt Connected InstrumentsInstruments

Thevinin

voltage

source oE

oZ

Output

impedance

LZ

Load

(instrument)

OnOn connectingconnecting shuntshunt connectedconnected

instrument,instrument, whosewhose impedanceimpedance isis ,,

thethe actualactual voltagevoltage decreasesdecreases toto

asas followsfollows::o

L

EI

+=

oE LE

LZ

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AndAnd thethe %%loadingloading errorerror inin measurementmeasurement

ofof equalsequals

LooL

Lo

oL

IZEE

ZZI

×−=

+=

oE

100E

EEerrorLoading%

o

oL ×−

=


AnAn oscilloscopeoscilloscope (CRO)(CRO) havinghaving anan outputoutput resistanceresistance ofof 11 MMΩΩΩΩΩΩΩΩ shuntedshunted

byby 5050pFpF capacitancecapacitance isis connectedconnected acrossacross aa circuitcircuit havinghaving anan outputoutput

resistanceresistance ofof 1010 kk ΩΩΩΩΩΩΩΩ.. IfIf thethe openopen circuitcircuit voltagevoltage hashas 11..00 VV--peakpeak forfor aa

sinusoidalsinusoidal ACAC--source,source, calculatecalculate %%loadingloading effecteffect errorerror ofof thethe voltagevoltage

measuredmeasured whenwhen frequencyfrequency isis:: ((ii)) 100100kHzkHz ,, (ii)(ii)11MHzMHz


1032jjX//RZ3ΩΩΩΩ××××−−−−≅≅≅≅−−−−====

27

%6.41000.1

0.1954.0errorLoading%

peakV4.17954.0

1032j10101

00.1

Z

Z1

EE

1032jjX//RZ

kHz100at

o

3

3

o

L

o

oL

3ckHz100atL

−−−−====××××−−−−

====

−−−−−−−−∠∠∠∠====

××××−−−−××××++++

∠∠∠∠====

++++====

ΩΩΩΩ××××−−−−≅≅≅≅−−−−====

Continue Solution Continue Solution 88::

%701000.1

0.1303.0errorLoading%

peakV72303.0

3183j10101

00.1

Z

Z1

EE

3183jjX//RZ

kHz100at

o

3

o

L

o

oL

cMHz1atL

−−−−====××××−−−−

====

−−−−−−−−∠∠∠∠====

−−−−××××++++

∠∠∠∠====

++++====

ΩΩΩΩ−−−−≅≅≅≅−−−−====

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%701000.1

errorLoading%kHz100at

−−−−====××××====


2.2. Loading Effects due to Loading Effects due to Series ConnectedSeries Connected InstrumentsInstruments

Thevinin

voltage

source oI

oZ

Output

impedance

OnOn connectingconnecting seriesseries connectedconnected

instrument,instrument, whosewhose impedanceimpedance isis ,,

thethe actualactual currentcurrent changeschanges toto

asas followsfollows::

oo ZE

oI LI

LZ

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AndAnd thethe %%loadingloading errorerror inin measurementmeasurement

ofof equalsequals

oLo

o

Lo

oL I

ZZ

Z

ZZ

EI ××××

++++====

++++====

oI

100I

IIerrorLoading%

o

oL ×−

=

Thevinin

voltage

source

LI

oZ

Output

impedance

LZ

Load

(instrument)

Uncertainty Analysis and Propagation of ErrorsUncertainty Analysis and Propagation of Errors

1.1. Errors Errors vs.vs. Uncertainty Uncertainty

i.i. UncertaintyUncertainty isis thethe rangerange thatthat isis likelylikely toto containcontain thethe deviationdeviation ofof thethe

measuredmeasured valuevalue fromfrom thethe truetrue valuevalue basedbased onon randomrandom--typetype errorserrors..

ii.ii. UncertaintyUncertainty cancan bebe expressedexpressed inin absoluteabsolute termsterms oror relativerelative terms,terms, justjust

asas errorerror cancan..

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For example, consider a meter stick that indicate centimeter and

millimeter divisions. Thus

1)1) TheThe smallestsmallest valuevalue youyou cancan readread onon thisthis metermeter stickstick isis 11 mmmm..

2)2) AnyAny measurementmeasurement youyou makemake willwill bebe quotedquoted toto thethe nearestnearest millimetermillimeter..

3)3) SoSo youryour answeranswer willwill taketake thethe formform 5555 mmmm ±±±±±±±± 11 mmmm ((5555 mmmm isis thethe

measuredmeasured valuevalue && 11 mmmm isis thethe rangerange ofof uncertainty)uncertainty)..


2.2. PropagationPropagation of Uncertainties of Uncertainties

i.i. AnyAny calculationscalculations donedone usingusing aa measurementmeasurement willwill havehave aa degreedegree ofof

uncertaintyuncertainty.. ThisThis uncertaintyuncertainty isis aa measuremeasure ofof howhow confidentconfident youyou areare inin

thethe resultresult ofof youryour calculationcalculation..

ii.ii. TheThe propagationpropagation ofof thethe uncertaintiesuncertainties throughthrough variousvarious calculationscalculations hashas

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toto bebe carefullycarefully consideredconsidered..


3.3. Propagation of UncertaintiesPropagation of Uncertainties: : Sum of Two or More QuantitiesSum of Two or More Quantities

yyxxwor

yxw

yxw

∆∆∆∆××××++++

∆∆∆∆××××====

∆∆∆∆

∆∆∆∆++++∆∆∆∆====∆∆∆∆∴∴∴∴

++++====

Let the final result w be the sum of the measured quantities (x ±±±± εεεεx) & (y ±±±± εεεεy)

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y

y

w

y

x

x

w

x

w

wor

∆∆∆∆××××++++

∆∆∆∆××××====

∆∆∆∆

Finally, the maximum calculation of w is

(((( ))))yw

yxw

xw εεεε××××++++εεεε××××±±±±====εεεε


4.4. Propagation of UncertaintiesPropagation of Uncertainties: : Difference of Two QuantitiesDifference of Two Quantities

yxw

yxw

∆∆∆∆∆∆∆∆∆∆∆∆

∆∆∆∆−−−−∆∆∆∆====∆∆∆∆∴∴∴∴

−−−−====

Let the final result w be the difference of the measured quantities

(x ±±±± εεεεx) & (y ±±±± εεεεy)

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y

y

w

y

x

x

w

x

w

wor

∆∆∆∆××××−−−−

∆∆∆∆××××====

∆∆∆∆


(((( ))))yw

yxw

xw εεεε××××++++εεεε××××±±±±====εεεε


5.5. Propagation of UncertaintiesPropagation of Uncertainties: : Product of Two or More QuantitiesProduct of Two or More Quantities

yieldswt.r.watingDifferenti

ylnxlnwln

yxw

++++====

××××====

Let the final result w be the product of the measured quantities

(x ±±±± εεεεx) & (y ±±±± εεεεy)

34

y

y

x

x

w

w

w

y

y

1

w

x

x

1

w

1

∂∂∂∂++++

∂∂∂∂====

∂∂∂∂

∂∂∂∂∂∂∂∂

××××++++∂∂∂∂∂∂∂∂

××××====


(((( ))))yxw εεεε++++εεεε±±±±====εεεε


6.6. Propagation of UncertaintiesPropagation of Uncertainties: : Division of Two QuantitiesDivision of Two Quantities


ylnxlnwln

y

xw

−−−−====

====

Let the final result w be the division of the measured quantities

(x ±±±± εεεεx) & (y ±±±± εεεεy)

35

y

y

x

x

w

w

w

y

y

1

w

x

x

1

w

1


∂∂∂∂−−−−

∂∂∂∂====

∂∂∂∂

∂∂∂∂∂∂∂∂

××××−−−−∂∂∂∂∂∂∂∂

××××====


(((( ))))yxw εεεε++++εεεε±±±±====εεεε


7.7. Propagation of UncertaintiesPropagation of Uncertainties: : w = f (x , y)w = f (x , y)

Using Root Sum Square (RSS) method, the error in w is defined by

(((( ))))22

2

y

wy

x

wxw

∂∂∂∂∂∂∂∂

××××∆∆∆∆++++

∂∂∂∂∂∂∂∂

××××∆∆∆∆====∆∆∆∆

36

(((( ))))2

21

22

21

yxyyxw

yxy

w&y

x

w

y

xwLet

23

21

23

21

−−−−××××∆∆∆∆++++

××××∆∆∆∆====∆∆∆∆

−−−−====∂∂∂∂∂∂∂∂

====∂∂∂∂∂∂∂∂

====

−−−−−−−−

−−−−−−−−

Example 9::

37

(((( ))))

(((( ))))2y212

x2w

2

21

22

21

22

2

21

y

y

x

x

yx

yxy

yx

yx

w

w

yieldswbysidesbothdividing

yxyyxw

21

23

21

21

22

εεεε××××++++εεεε====εεεε∴∴∴∴

∆∆∆∆++++

∆∆∆∆====

××××

−−−−××××∆∆∆∆++++

××××

××××∆∆∆∆====

∆∆∆∆

−−−−××××∆∆∆∆++++

××××∆∆∆∆====∆∆∆∆

−−−−

−−−−

−−−−

−−−−

electrical & electronics measurement ee302 · 2012-06-27 · loading effects ii.. underunder...

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