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1. STATIC CHARACTERISTICS OF INSTRUMENTS AND MEASUREMENT SYSTEMS

247 ERRORS IN MEASUREMENTS
New

‘Since errors are a must in er

at tobe measured may be defined as he average of an infinie number of
on nun rage deviation due tothe various contributing factors tens 10 zero, Such an
‘del station is possible to realise in practice and hence itis not possible o determine the "True Value”
oh gangs RY Serine ment. The reson forts Ts Uat e pose deviations frm the
ie value do not equal the negative deviations and hence do not cancel each other. Thus, normally an
experimenter would never know ence d À each eter. Thus, normally

at the value of quantity being measured by experimental means is the
Sue Value” ofthe ant orage Samy bin measured by experimental °

An fact in practice, the tenn “True Value", then refers 0 value that would be obtained ifthe quantity
‘consideration were measured by an “Examplar Method”, Dat is a method agreed upon by experts
as being sufficiently accurate forthe purposes to which the data will ultimately be put to we

2§/ STATIC ERROR

‘The most important characteristic of an instrument or measurement sytem is its accuracy, which is
the agreement of the instrument reading with the true value of quantity being measured. The accuracy of
an instrument is measured in terms of serrer.

‘We have mentioned earlier tha its impossible to measure the te vale ofa quantity. An approximation
ofthe “true value” obtained by sufficiently extended series o measurements and also taking into account
parameters and conditions to which coretions may be applied, we obtain, what i called the best measured
value ofthe quantity. While it i never possible o measure the true or exact vale of a quantiy, ts nazly

always possible to give a best measured value, Stati error is defined 25 the difference between the measured
value and the true value of the quantity. Then

Sf 7 a”
where 5
“A = measured value of quantity,
and À = tre value of quantity
{A is also called the absolute state error of quant} A.
We have Der en

where ey = absolute satie error of quanliy A (under measurement)

“tn absolute value of 84 does no indicate precisely the accuracy of measurements. As an example,
an error of 2 2 minis negligible when te length being measured is ofthe order of 1 m (1000 mm) while
or Of € 2 mm may be regarded as intolerable when the length under measurement is 10 mm
a sas the quality of measurement fs provided by tho relative static error, Le the ratio of absolut
Sore res SA to Ay the ue value of the quanity under measurement, Therefore, the relative static
error €, is given by

absolute cor BA So,
SE “te value = À = Ay es
Bene 24)
Percentage static error we e,=6,% 100 04)

2. D INSTRUM,
QUA MEASUREMENTS ANI oy
ecu

= (see Eqn. 2.1)
Voie © (ee Egns. 22, 23)
es
A EE When the absolute static error € = 3 is small,
‘ed and tue values i mal e 26
Fan. 25 may be writen as: 4, = A €)
STATIC correction
a lue of the quantity, or
"te eng en te vals a end vaa de Sa
Free »
where

=- 64. 428)
C= se carton
In order to emphasize the N of relative error and to ensure that this idea of relative ‘size ig
"emt ini, he owing ena a

cate ofa pera mcr he temperate ofa face which is 165070 and ie
then the pecs elaine

5
ern x 10022 0300 porent
Teno 100 £0308 ps

However if he ane enor de, 45% acu in the measurement of temperature of boiling wate at
00°C, then the relative pare a is,

Static correction i
Example 22. A thermometer eas 980
OBC. Detemine the te vale of eer
Solution. True value ofthe tempera A4 + à
623. A vole has au eof à
270-250 V shows a voa of AV Wa
To as fai ofthe ue valo and the sa
Solution. Absolute error Asa
Absolute correction äc =

EEE ing num
Va of shot a a

nt With a scale range
Sec (ag "24 orton, eae

Hn 146130094 y

Relative error e

ample 24. A cd manom u mens
Bean este en Pre Pow ae ads 0.61. 10-1
ms. HR Pi (0 ac em

Solution. (a) Pom Bar. 21, th ie exo y and () static correction,

As (0161 9 159) x 10-32 0.997 X 10 me,

a ee IA | = |

3. (ATC CHARACTERISTICS OF INSTRUMENTS A
= IMENTS AND MEASUREMENT SYSTEMS >
(0), Static correction

8C =~ 84 =~ 0002 x 10-3 ms.
ALE RANGE AND SCALE SPAN

inem eS Hude, =

Fee goin rine dd ee

sro net nd omg ef teu Si ga ces

beta (5 Xi units and that the calibration is continuous between the two points.

ee ring sn in du
The tent ren by San ac, ES
The sme itn spy nth ee of il en

For a pyrometer calibrated between 0 o 1000°C, th PC. Fe
o 1000°C, the range is 1000°C and span is also 1000°C. For
thermometer calibrated between 200°C to 500°C, the range is 200°C to 500°C (or SOOPC) but the span

1500-200 = 300°C. For a thermometer having a cairalio of = 40°C 10 + 400°C the rane is 400°C
and the span is 400 ~ ( - 40) = 440°C, de sl

There is another factor that must be considered while determining th range of the instrument, This
isthe Frequency Range, which is defined as frequencies over which measurements can be performed with
specified degree of accuracy. For example a moving ion instrument may have a0-250 V range and 0-135 Hz
frequency range with an accuracy of OS or 19% of full scale reading.

Example 25, A thermometers calibrated 150°C 10 200°C. The accuracy is specified within + 025
percent of instrument span. What isthe maximum static emor 7

Solution. Span of thermometer = 200 ~ 150 = 50° C.

2, Maximum static error

2 025 x &

100

RO CALIBRATION CURVE
Emursalvaonmensinan +2
im a ben ale * T
mat sale te 3
Sein re 22 an se
ror demie a à umber el
ins on sea esd om
Eero curve, whichean be se fr
Es men ro À
te ororcuneie shorn PE
E

ous 0.

1

|
In this context, we cannot but
overemphasise the importance of
calibration which firmly establishes |
the absolute accuracy of an
instrument or a measurement system,
We should not accept the readings of

a

Static error, percent

mai EE CE EE
clon Meosured variable , percent span
e sar of validity of i.

4. MECHANICAL MEASUREMENTS An INSTRUM
2

Ug,
AU SCALE RADA

aa instruments pati bys human ¿server noting the position q
gra cartel set

Do
Th a fo am eye amo mae St me
the closeness with wie

isi

on a e ead. is abo i
le scale of an analog (ype Oltra ms
shan of 200 mm wut ave higher readability than da instrument having a scale span or 100 mm aa
Hey readability than gr bier
ber of graduations, spacin radio,
bi an cen
mae ae in as
he a higher namie

D of the eae oF a

ame input is applied repetitively
me and ber ea

rh TSE e e
suring ‘instrument, locatiog Sonditions of use and time of

ic ein anni at
a Den na na tte
called Drift No an ire a th ge

‘ay be classed fc Eos une aa
ire te Dei. ihe in

warming up ote

‘ouput characte

rit or eit ae ae Br
ER Dri o Senta Dit. tr there Propo
Mn ge de dit ate i a sea to “HOI chang ig
1” Fit. 22 () and nen wh oth zero an tls The ex

mn a

Characteristics Choracterigt

m eis efoctriti
E ME ca

NN :

27 cRaraetetistcg lenin
A, tie,
=
F Input Mar —

19 Coatings

5. ATC CHARACTERISTICS OF INSTRUMENTS An

sate en men ih y my ty dc mate

is an undesirable quan
ay campers for Ths any nd inroment Bao ely apparent and cnn
an st cry gue sin hy oman even secte ad
mai + say elecrostaic and electromagnetic fields can be prevented fom affecting
the measurements by proper shielding, Effect y O as ore
proper shielding, Effect of mechanical vibrations cn be minimized by having proper

moumings. Temperature changes during the mensurement pro
te properly compensated foe proces shouldbe preferably avoided of otherwise

sample 26. À il gauge wid to ment presse in veel, The pivot is ck xe at he
cs andas elt of which height er as fn athe pation
makes ng 00 ars in à linear fashion, The readings are 695 Km! fr a dial reading of 270
and 27.58 N/m? for a reading of 150. What would be he value of pressure for a dial reading of 100 ?

Solution. This is a case of zero drift, The dia reading i offet by 6895 kN/m? at zero pressure and
reading at 150 indication of dial is 27.58 KN/m?,Sine the variation i linear, the indication at dil reading
of 100 is

a cant tg cnt cin ge is ie ro
seo als Nae Tt nana vena
Seapine grees ;
forms related o vital human organ in biomedical instrumentation systems ‘where the expected (desired)
See al ca wir spa neo
ed nem el nein eh
fe oe om eget ie Te oye of Bc
pers
5 signal power

11 the unwanted nis is called signal to noise ratio and is expressed as

sal of interest expressed in VOD ‘ait
unwanted noise expressed in vol?

6. 3 MECIMNEA, MEASUREMENT AND u,
ta

day nssremen sem iis deal o ave resign toast. This can be
by inerenong th siga! level witht increasing th moi eve or decresing Uh noise level a
Stable tchnique

2182 Sources of Noise, The nie out
in Abe oops oly of the opos: (0) tout
enerated noise. Ci) conducted. oise, Amplifier Signal nak
(Gi radiated noise. Th can be lasted by

Sonsidering the case ef an amplifier which is
‘commonly used in measurement systems lo
entity the different sources of noise, The
diagram of the amplifier is shown in

Power ‘Supply
Generated Noise, Suppose the
can SENO] Coninins no noise. The power Pl 23 an amplio rang te ern peso ace ip,
Tens Serve à a source of enery fr ih operation of ampli, The ouput signal is amplifier gain mes
ifs saat sien plas a oise signal, On ofthe possible sources oie sn account o intel compone
AS exo, capacitors and transistors ci, Therefore the noise in this ease fs generated
and therefore i called Generated nose. The internally generated noise ison account
‘of Somponents like resistors, capacitors, transistors ei. a sated above
“© conductive portion of a resistor consists ofa regularly arranged-groups of atoms that maintain
nee physical postin inthe conductor. These atoms contribute conduction electrons dus ty
ich a cureo: fiews. Although the atoms maintain thei general physical postion, they are inva ae of
rapid vibratory motion on account of temperature and thermal eects, This vibratory motion of ana Le
transfers 10 ths conduction elecuons, thereby producing nise component of cuen, Since ie nei
erature dependent it increases with internal heating (PR ls) or with an increase ln he an
ie is called Johnson noise,
‘Thc vibrations produced by bemal effects within a resistor cover a wie frequency range. and therefore
< generated consists of a wide spcrum of frequence. This wideband noises sometimes cali

"perature. Th

White noise.
“hry ected in esr cbr y cio siti opera Spa

film and las eubsrats are wed for minimising is ne
As nos nera eerie, tal sig cmo bp in ig hs o Alo se
the mies a wi een amd lie in 1 lie rece A in
‘sez o of notes eid nly er me eel o ag li
ikea tans In vn concn devs ce otis or o pn ¿es
ee ons omc el ane Tene oe a vi de
ich generate eecuomapneti stan, which nun pace nia por ES
% cd of uawsition fram one energy level 10 another is short, the period ‘of acceleration is also short,
infest en pe wae Tsay bs des
23 ern pss gh vaso len sea ofa ac be a Ed
Me imenall goad ion sensual and taa be dees on aout of random
The jotemall 8 "Noise. Not much can b dono to reduc he incre ponente none

iy of charges i called Sh

mover ngs an vane devs sis ae hon can fenton robe

ie els in he ion betwen plates of spiro canine maga els rnd

ú surces of imermolly generated noise, However, in these cases the electric and magnetic

Care ae sa

7. rare CHARACTERISTICS OF INSTRUMENTS AND MEASUREMENT sy ”
“Thus, when it is stated that a set of à

idings shows precision it 15 that the results agree amont
ihemselves. Agreement, however, is no guaran e anos Ss
antes Ann, ween pra ym see omnes
30 INDICATIONS OF PRECISION

(6) Conformity and (i) Number of sign

o : cigs.

eco ed in measurements o des men athe epee of esas. A quality
cao pression inde ied im ap on age cio rer sario of te esa
St same cera vale. High ein mean sgt histo eps esl whl preci nen
br setting of resus But ths may not lead st e misconception hat high precision estes high
Segre of accuracy since al the repetiion in rest may be ein he same wy by some senal
tet th produce sare devin fret om het ve For example. a sping ele Usd vi
Spring designed fora cern sping cle may repeal show ie same vale al weight. Thus raiz

may display exceptional agreement between themselves, ut they all would be inaccurts values of weight
On account of use of a wrong spring introduces à system

‘shift of all readings

‘We may well cite another example o ustate another spect of precision. Consider, fo
hat a resistor whose true resistance I, 1.385.692 Q, is being measured by an ohmmeter, The ohm
consistently and repeatedly indicates the tru valu. But the observer cannot read his value from the sele

‘The observer's estimates from the sale reading consistently yield a value of 1.4 MQ. This is as close to
the true as he can read the scale by estimation. Although there are no deviations from the observed value
the error created by the imitation of the sale reading is a precision error.

"The above example ilustrats that conformity is necessary, but not sufficient
because of lack of significant figures obtained Similar, precision is necessary, but a
for accuracy

a work, good patie requires an independent set of measurement, using diferent instruments
oe ee nen techniques which are not suje fo the same sptematc emos. Where his is no
are epetimener mus ake sep tains rope anioning of inmensa 1 isonet sl
epale curbing cos Cain agaist a own sand may scort inorder
to achieve the above purpose.

249/ SIGNIFICANT FIGURES

E on of ing pression of the measurement bain rom he ui of ica tes
sawn neti A giant gues convey acta ifomation eating the moze ae
em pression of a quant. The more the signi

dition or precision
sfliien condition

| ke N 97255 Vi valo of vole deve a 2860 Y. means I he a

8. A nn NO
lse can large numbers be
0 significant figures. How el > pre

te ny oe pl a ete x10 o 49 CIO, Tas a zu
‘Would be interpreted automatically as an approximate number while ref ee 10 velocity of tig A
ttn gan age ua
the velocity flight 3 x 108 mer er second eaves no scope for oo e Ty eased by ot
{0 the left of decimal point is therefore usually resolved by scientific nc ME POWers of e,

Example 2.11. State the number of significant figures in each of the following numbers.

(a) 302 m (b) 302.10 °C (c) 0.00030 0 (d) 0.000030 MPa (e) 5.01 x 10 502000,

Solution. (a) The number is 302, This means that it is more close 10 302 than to ether 301 6 30,
Tsismis ha pan tae

(e une ei BE e im cl 3210 ed 1
ssh sl ae

u Tis timer oO dani 0 etn,

[O De rca 3 0 MP ran win 30 Tis mae do eg

Seo eran 739731 Pa Teer ha 2 sign gue. The 21081 he len at See ol
size of ui

Population is close to 502 x 10 dan

Si SO 1058 10% Ti man a amis a furs,

ve kon 2.
as u en etn
6.A similar but definite method ist express te ab Wer as 226 4 005 Wines thatthe ponet
fies between 7255 W and 2265 W gg lago of se
When me nn 3
Mee ly ie al a
is the largest deviation from the mean, readings. Te
mp2 À et em, a
1008 N, Ce 0) tro and (mag real 1003, 100,100
Solo. .

1293 + 1010 + 10.1 + 1008
AAA hn:

OU N,

CITAN
(a) Average force F, = LERMA

(6) Maximum vale of force E,

9. 2 MECHANICAL MEASUREMENTS AND, sn]

Solution. (leo significant figures)

an
¿tour significant figures)

min
287 + 2004
- 2.040
9030
This is done Because one of the resistaces is acute only 1 thee significant figures (or oy,
a ohm in bi exe) and therefore heul ld lso he ede ee significant Figures (rhea

tenth of an ohm in this case) and hence the value 32.3 62
‘Te nuke of significant figures in multiplican may increase rapidly, but only the appropri fu,

are retained in the answer, ac ie lutte hy the following example
Example 216. In calculating volag drop, a current of 4.37 Ais recorded in a resistance of 3127 q.

Calculate the voltage drop across the resistor tothe appropriate number of significant figures
Solotion.

‘Ton resistance
“ (ive significant figures)

three significant figures)

Current 2 = 47 À (thre significant figure)
Resistance R= 31.27 2 (four significant figure
Voltage drop E = IR = 437 x 3127 = 1366499 V (seven significant figure
Since thre are three significant igus involved in the multiplication, the answer can be writen ony
10 3 maximum of thee significant figures ie, E = 137 Y.

24

Tie em “dsr” ness sme exp des ly dsd meaning. he ii
cece Between mas va o er et parco à pl lc
Satish ee ln des e A lr pl a a) a et
chained two dren slo sm gsi, ice ha hee usd E

239/ TATI SENSITIVITY
aa pao : e natant ae
‘of micro-empere pes as the cae may be depending upon nature of

factor or inverse ss, Many mana den he ents in terms of
verse sensitivity and sll ll sally, runes te

cabanon curse à near as

When in Fig 2.8 (a) the sensitivity of can be defined
murument. Howene, übe Sue nominal a ira in, the som a ange of e

10. ECIANICAL MEASUREMENTS AND STR,
ha,

“4 i
ange in it ara and lng,

on material and hence there will be no e
the bulb has a zero expansion material

de orale Mas mayo

Now A, (La + la) = Ae (a+ 0, Le 20)

Leng of expiry tbe
CN Las
six 103

Hence aca of bulb Ay = AL = Ÿ (025) x 138 x 10° = 6774 mm.

138 x 10° mm = 138 m.

LINEARITY
One fie best characteristics of an instrument or a measurement system is considered t be liner,

that the ouput is lineal proportional o the input. Most of the systems requir a linear behaviour a,
it is desirable. This is because the conversion from a scale reading to the corresponding measured vl
of input quantity is most convenient if one merely has to multiply by a fixed constant rather than consu
‘a non-linear calibration curve or compute from non-linear calibration equations. Also when the instrument
is part ofa age data or contol system, linear behaviour of the par often simplifies the design and analysis
ofthe whole system. Therefore relationships to the degree of straight line relationship (ie, linearity) sre

If for an instrument calibration curve (relating output o input) is not a straight line, it should not be
concluded thatthe instruments inaccurte Ti ia misconception, non near behaviour doesnot essential
lead to inaccuracy. Such an instrument may be highly accurate 35 ever.

However, most ofthe time it is necessary hat measurement system components should have ine
Charcterisies. For example, the resistance used in a potentiometer should vary linearly with displacement
‘ofthe siding contact in order that so that he sliding contact voltage is directly proportional othe displacement.
‘Any departure from linearity would resul in enor in the read out systems

‘There are many definitions of linearity that exis, However, linearity defined in terms of Independent
Laneros mstefrbi many cas. he compan nero ny doo eee

o a stright line showing de
relationship. between input and
output. This straight ie is drawn by
Using the method of least SAS
from the given calibration dats Tis
Actual + Staight line is sometimes called 2
calibration + idealized straight ine expressing te

E cure
3 ds input-owput relationship. The
ii linearity is simply a measure
maximum deviation of any ol 1%

calibration points from this sight

line. Fig. 26 shows te acual

calibration curve e. a rela

between input and output ad 2

Man — he rn ro de rg
Fig, 26. Linent depicts ion teen coc ing the method of es saure
O independent sone

may be defined