M&I-203-Errors

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Measurement Errors

Measurement Errors

Ebrahim A. Badran Spring 20131

Ebrahim A. BadranPh.D., IEEE Member

[email protected]

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Measurement Errors

Introduction

Measurement is essentially the act, or the result, of a quantitative comparisonbetween a given quantity and a quantity of the same kind chosen as a unit anda measuring instrument is a device used for comparing the unknown quantitywith the unit of measurement or a standard quantity.


e va ue o e un nown quan y o a ne on ma ng measuremen s wstandards and measuring instruments is considered to be its true value, thoughpractically it is never.

There is always some difference between the measured value and the true orexact value of the unknown quantity.

The difference between the measured value A m and the true value A of theunknown quantity is known as the absolute error of measurement , A,

A = Am - A

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Measurement Errors

Introduction

The absolute value of error A does not indicate precisely the accuracy ofmeasurements.

The quality of measurement is indicated preferably in terms of relative error.


be measured.

When the absolute error 0=A is negligible, i.e., when the difference betweenmeasured value A m and true value A is negligible then relative error may beexpressed

TrueValueror AbsoluteEr

A A A

r lativeErro or ===

,Re

mr A

A =

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Measurement Errors

Introduction

The relative error may be quotted as a fraction e.g., 5 parts in 1000 or may beexpressed as a percentage

%100%100 == A

Error Percentage or


The amount of relative error is often an indication of the class of use to which apiece of apparatus may be put.

For example, a resistor which is within 5% of its nominal value may be quitesatisfactory in a shunt field regulator or in a radio set but useless as a standard

or in a decade resistance box.A resistor within 1% of its nominal value can be quite useful for generalmeasurements but not for very precise measurements.

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Measurement Errors

Introduction

Circuit components such as resistors, inductors and capacitors are guaranteedto be within a certain percentage of the rated value where as in indicatinginstruments the accuracy is mostly guaranteed to be within a certain percentageof full-scale reading.


of a particular quantity in order to enable the purchaser to make proper selectionaccording to his requirement.

The limits of these deviations from specified values are defined as limiting or guarantee errors.

For example, if the resistance of a resistor is given as 8005% the manufacturerguarantees that resistance falls between the limits of (80040) , i.e., between760 and 840 .

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Measurement Errors

Introduction

The magnitude of a given quantity having a measured value Am

and a maximumor a limiting error A must have a magnitude between the limits

Am - A and A m+ A


, = m

For example, the measured value of a capacitor is 100 F with a limiting error of5 F.

The true value of the capacitor will be between the limits 1005, i.e. between 95

and 105 F.

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Measurement Errors

Example 1.

The measured value of a resistance is 10.25 , where as its value is 10.22 .Determine the absolute error of measurement.

Example 2.


A wattmeter reads 26.34 watts. The absolute error in the measurement is -0.11watt. Determine the true value of power.

Example 3.

The measured value of a capacitor is 205.3 F, where as its true value is 201.4F. Determine the relative error.

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Measurement Errors

Example 4.

The inductance of an inductor is specified as 20 H 5 % by a manufacturer.Determine the limits of inductance between which it is guaranteed.

Exam le 5.


A 0-25 A ammeter has a guaranteed accuracy of 1 % of full-scale reading. Thecurrent measured by this instrument is 10 A. Determine the limiting error inpercentage.

Example 6.

A 0-250 V voltmeter has a guaranteed accuracy of 2% of full-scale reading. Thevoltage measured by the voltmeter is 150 volts. Determine the limiting error in

percentage.

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Measurement Errors

Resolution And Sensitivity

If the input is slowly increased from some arbitrary value it will be noticed that theoutput does not change at all until the increment exceeds the certain value calledthe resolution or discrimination of the instrument.

Thus the resolution or discrimination of any instrument is the smallest change in


instrument.

It may be expressed as an actual value or as a fraction or percentage of the full-scale value.

In some texts this quantity is referred to as 'sensitivity'.

The sensitivity gives the relationship between the input signal to an instrument ora part of an instrument system and the output.

The sensitivity is defined as the ratio of output signal or response of the

instrument to a change of input signal or the quantity under measurement.

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Measurement Errors

Example 10.

A moving coil voltmeter has a uniform scale with 100 divisions and gives full-scale reading of 200 V. The instrument can read up to 1/5 th of a scale divisionwith a fair degree of certainty. Determine the resolution of the instrument involt.


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Measurement Errors

Accuracy And Precision

Accuracy is a closeness with which the instrument reading approaches the truevalue of the variable under measurement while precision is a measure of thereproducibility of the measurement i.e., precision is a measure of the degree towhich successive measurements differ from one another.


,defined as sharply or closely defined.

Precision is composed of two characteristics; conformity and the number ofsignificant figures to which a measurement may be made.

Consider an example in which a voltage reading of true value 1.952 V ismeasured by a voltmeter which consistently and repeatedly indicates 2 V.

This is as close to the true value as an observer can read the scale byestimation.

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Measurement Errors


Although there are no deviations from the observed value, the error caused bythe limitation of the scale reading is a precision error.

So conformity is a necessary but not sufficient condition for precision, becauseof lack of significant figures obtained. Similarly precision is a necessary but not


su c en con on or accuracy as exp a ne e ow.

If a magnitude is to be determined with accuracy to a required number of digits,it is necessary that the measuring instrument have precision of this order.

Thus precision is an essential condition for achievement of the accuracy.Accuracy is a matter of careful measurement in terms of an accurately knownstandard.

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Measurement Errors


If the results of measurement of the same variable agree among themselvesthen that set of readings shows precision but it does not guarantee the accuracyas there may be some systematic disturbing effects which introduce error in thereadings.


, ,

by second method is also taken to avoid the effects of some systematic errors,where this is not possible to apply second method then method under useshould be studied carefully to find out and eliminate and systematic disturbingerror.

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Measurement Errors

Significant Figures

.


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Measurement Errors

Types Of Errors

No measurement can be carried out with complete accuracy, so h study of errorsis necessary in the study of measuring process.

A study of errors is a first step in finding ways to reduce them and such a studyalso allows ;o determine accuracy of the final test results.


To study the matter of accuracy, it is very important to know the different types oferrors that may enter during measurement of any quantity.

Errors may originate in a variety of ways, but it can be grouped in three maincategories.

1. Gross Errors2. Systematic Errors3. Random (or Accidental) Errors

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Measurement Errors

Types Of Errors

1- Gross Errors.

In this category, errors occur because of mistakes in reading or using instrumentsand in recording and calculating measurement results.


magnitude and cannot be subjected to mathematical treatment.One common gross error frequently committed during measurement is improper use of measuring instrument.

Any indicating instrument changes conditions to some extent when connected into a complete circuit so that the reading of measured quantity is altered by the method used.

For example in figs. 2.1(a) and 2.1(b) two possible connections of pressure coiland current coil of wattmeter are shown.

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Measurement Errors

Types Of Errors

1- Gross Errors.


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Types Of Errors

1- Gross Errors.

The connection shown in fig. 2.1(a) is used when applied voltage is high andcurrent flowing in the circuit is low while the connection shown in fig. 2.1(b) isused when applied voltage is low and current flowing in the circuit is high.


If these connections of wattmeter are used in opposite order then error is liable toenter in wattmeter readings.

Another example of this type of error is in use of a well calibrated voltmeter for themeasurement of voltage across a resistance of very high value.

The same voltmeter, when connected in a low resistance circuit may give a moredependable reading.

This shows that the voltmeter has a loading effect on the circuit, which alters theoriginal situation during the measurement.

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Measurement Errors

Types Of Errors

1- Gross Errors.

As mentioned above that these errors cannot be treated mathematically so agreat care should be taken during reading and recording the data to avoid theseerrors.


Apart from this, at least two, three or more readings should be taken of thequantity under measurement, preferably by a different observer at differentreading points.

Then if the readings differ by an unreasonably large amount, the situation can beinvestigated and the had reading eliminated.

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Measurement Errors

Types Of Errors

2- Systematic Errors.

Such errors arc errors that remain constant or change according to a definite law on repeated measurement of the given quantity.

These errors can be evaluated and their influence on the results of measurement


can be eliminated by the introduction of proper corrections.

These errors are sometimes referred to as bias, and they influence all measurements of a quantity alike.

A constant uniform deviation of the operation of an instrument is known as a

systematic error. There are basically three types of systematic errors:

(a) Instrumental, (b) Environmental, and (c) Observational.

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Measurement Errors

Types Of Errors

2- Systematic Errors. (a) Instrumental Errors.

Such errors are inherent in the measuring instruments because of theirmechanical structure and calibration or operation of the apparatus used.

For exam le in the DArsonval movement friction in bearin s of various


components may cause incorrect readings.Improper zero adjustment has a similar effect. Poor construction, irregular springtensions, variations in the air gap may also cause instrumental errors.

Calibration errors may also result in the instrument reading either being too low ortoo high.

These errors may be detected by checking for erratic behaviour, stability andreproducibility of results.

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Measurement Errors

Types Of Errors

2- Systematic Errors. (a) Instrumental Errors.

A quick and easy way to check an instrument is to compare it with anotherinstrument of the same characteristics or compare it with a comparatively moreaccurate instrument.


Such errors may be avoided by:(a) selecting a proper measuring device for the particular application,(b) applying correction factors after determining the magnitude of instrumental

errors, and(c) calibrating the measuring device or instrument against a standard.

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Measurement Errors

Types Of Errors

2- Systematic Errors. (b) Environmental Errors.

Such errors are much more troublesome as these change with time in anunpredictable manner.


ese errors are n ro uce ue o use o an ns rumen n eren con onsthan in which it was assembled and calibrated.

Change in temperature is the major cause of such errors as temperature affectsthe properties of materials in many ways, including dimensions, resistivity, springeffect and others.

Other environmental changes also affect the results given by the instrumentssuch as humidity, altitude, earth's magnetic field, gravity, stray electrostatic andmagnetic fields etc.

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Measurement Errors

Types Of Errors


These errors can be eliminated or reduced by taking the following precautions,

(i) Using instrument in controlled conditions of pressure, temperature and


,

(ii) If above is not possible then deviations in local conditions must bedetermined and suitable corrections to instrument readings applied,

(iii) Using equipment which is immune to these effects. For example, variation inresistance with temperature can be minimized by using resistance materialsof very low resistance temperature coefficient.

(iv) Using techniques that eliminate the effects of these disturbances. Forexample the effect of humidity, dust etc., can be entirely eliminated byhermetically sealing the equipment.

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Measurement Errors

Types Of Errors


These errors can be eliminated or reduced by taking the following precautions,

(v) The effects of external electrostatic or magnetic fields can be avoided by


,

(vi) Altogether new calibrations may be made in the new conditions.

Such errors are introduced by the observer. The most common error is the

parallax error introduced in reading a meter scale, and the error of estimationwhen obtaining a reading from a meter scale.

Parallax error is caused by the observer not having his line of sight on the pointerexactly at right angles to the plane of the scale.

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Measurement Errors

Types Of Errors


Such an error can be eliminated by providing a mirror beneath the scale and aknife-edged pointer.


es s ave een con uc e n w c a num er o persons rea rac onadivisions on an ammeter scale under carefully controlled conditions andconsistent individual differences were found.

In measurements involving the timing of an event, one observer may tend toanticipate the signal and read too soon.

Very considerable differences are likely to appear in determination of lightintensities or sound levels.

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Measurement Errors

Types Of Errors

2- Systematic Errors.

Systematic errors can also be subdivided into:Static errors, or Dynamic errors .


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Measurement Errors

Types Of Errors

2- Systematic Errors. - Dynamic errors

Dynamic errors are caused by the instruments not responding fast enough tofollow the variations in a measured variable.


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Measurement Errors

Types Of Errors

3- Random (or Accidental) Errors.

These errors are of variable magnitude and sign and do not obey any known law.

The presence of random errors becomes evident when different results are


.

The effect of random errors is minimized by measuring the given quantity many times under the same conditions and calculating the arithmetical mean of the values obtained.

The mean value can justly be considered as the most probable value of themeasured quantity since random errors of equal magnitude but opposite sign arcof approximately equal occurrence when making a great number ofmeasurements.

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Measurement Errors

Types Of Errors

3- Random (or Accidental) Errors.

The problem of random errors is treated mathematically as one of the probabilityand statistics, and is beyond the scope of this book.


e errors are rue y ran om, a p o o e num er o rea ngs o eac va ue asordinates, against the reading values as abscisae gives a curve of a particularshape known as the error curve.

If the plot does not give such a curve then the errors arc not truely random.

This test cannot of course be applied to a set of as few half a dozen.

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Measurement Errors

Determination Of Maximum Systematic Error

When two or more quantities (each is subject to error) are combined, it is

necessary to determine the maximum systematic error.It should be noted that when the information about the errors of two apparatus isthat they are within certain limits, they may be anywhere within these limits up to


.

In calculating the maximum possible systematic error it is necessary to assumethat the individual errors connected may all be of such a sense as to affect theresult in the same direction.

Provided that the errors are small, their effect on the final result is readilyobtained from the simple rules :

Difference of Two QuantitiesSum of Two or More Quantities

Power of a FactorProduct of Two or More Quantities

Composite FactorsQuotient of Two Quantities

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1-Sum of Two or More Quantities

It should be effect on the final result is readily obtained from the simple rules :


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Measurement Errors


3- Product of Two or More Quantities

It should be noted that when the information about the errors of two apparatus isthat they are within certain limits, they may be anywhere within these limits up tothe are small, their effect on the final result is readily obtained from the simple


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Measurement Errors


4- Quotient of Two Quantities

It should be noted that when the information about the errors of two apparatus isthat they are within certain limits, they may be anywhere within these limits up tothe are small, their effect on the final result is readily obtained from the simple


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M E

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Measurement Errors


6- Composite Factors

It should be noted that when the information


M t E

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6- Composite Factors

It should be noted that when the information


Measurement Errors

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Measurement Errors

Example 11.

Two capacitor* IfiO * X.4 HF "* IM * l* aF ara conn.ni.d In paralUl.DUrmIn#

(In- limiting orror of the rnaultant cnpoU*nri< In |*K and la p*ro#n ! ;..


Example 12.

Thrao re*l*tar* hv ihn following rating*;It, 200 O * 6%, K, - 100 U a ft*. K, DO * 6% U*t*rmln tb* magnitude of

r**ultant r"l*tanca And limiting *rror In porcoutag* aad ohm* If ilift ohovormuaticin or* eonntetad In *rlH and (b) parallol.

Measurement Errors

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Measurement Errors

Example 13.

The unknown resistance is determined by Wheatstone bridge and is given bythe expressionR = R2_ x Rl*where R, = 100 * 0.4% 1; R x = 1,000 * 1% O and R, = 1,100 0.6% aDetermine the magnitude of the unknown resistance and limiting errors in per


cent..

Example 14.

An unknown roililnnco in dwiormintfd by Whiuilnlim* bildf mid l given by lb*

*praailonR - a"5iwhfru K, - 100 * 0.4* Q( H, NH> * 0.7"*. nd K 3 HIS * 0.5%.pfftrfhilna tho niatfnluidfl or ibv unknown ri*tncn mid limiting arrori In por Otal*nd In

Measurement Errors

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Measurement Errors

Statistical AnalysisNo measurement is made with 100 per cent accuracy and, therefore, there ualways some error, which varies from one determination to another, und getsintroduced in the value of the quantity under measurement. It in a function afstatistics to separata, an far a possible, the truth from error by meowing unddefining the region of doubt. But statistical study in mainly concerned with


data. So systematic errors Ahuuld ho small un compared with residual or randomerrors,To make statistical methods and interpretations meaningful, a large number ofmeasurements la usually required.Sometimes simpio approach is required for describing and summarising the

results of the measurements. Some of thoao methods are described below.

Measurement Errors

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Measurement Errors


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