Chapter 0010

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Measurement Errors

The measurement of any quantity plays very important role not only in science but in

all branches of engineering, medicine and in almost all the human day to day activities.

The technology of measurement is the base of advancement of science. The role of science

and engineering is to discover the new phenomena, new relationships, the laws of nature

and to apply these discoveries to human as well as other scientific needs. The science and

engineering is also responsible for the design of new equipments. The operation, controland the maintenance of such equipments and the processes is also one of the important

functions of the science and engineering branches. All these activities are based on the

proper measurement and recording of physical, chemical, mechanical, optical and many

other types of parameters.

The measurement means, to monitor a process or a operation and using an instrument,

express the parameter, quantity or a variable in terms of meaningful numbers. Such a

measurement gives in depth knowledge of the process and the parameter and helps in

further modifications, if required. Thus the measurement provides us with a means of

expressing a natural phenomena or the various processes, in quantitative terms. The

feedback information is possible with the help of measurement techniques, which helps in

achieving goals and objectives of various engineering processes and systems.

The measurement of a given parameter or quantity is the act or result of a quantitative

comparison between a predefined standard and an unknown quantity to be measured.

For the result to be meaningful, there are two basic requirements :-

1. The comparison standard is accurately defined and commonly accepted, and

2. The procedure and the instrument used for obtaining the comparison must be

provable.

The major problem with any measuring instrument is the error. Hence, it is necessary

to select the appropriate measuring instrument and measurement procedure which

minimises the error. The measuring instrument should not affect the quantity to be

measured.

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This is the actual value of the unknown resistance.

At - A 5.462 -5.333ii i ) % error = At m X 100 5.462 x 100

= 2.36 %

iv ) The relative accuracy,

% A (1 -Ierrorl) x lO O = (1 - 0.0236) x lO O

97.63 %

1 . W ha t is mea surem ent? W hat ar e the two basic requirements o f an y measurement?

2. List the advantages of an elec tronic measurement.

3. De fine and cxplain tilc t erm 'Calibration '.

- t . How the performance characteristics o f a n i nstrumcnt ar e classified?

5. Definc and e xplain the following static characteristics of an instrulllcnt :

i) Accuracy ii) Prec ision iii) Static error iv) Resolution

v) Sen sitiVity v/) Th reshold vii) Z er o drift viii ) Rcp rodu cibility

ix) Lillearit y a nd x) Stability

6. Exp lain how the accuracy can be spec ified for a n instm ment.

7. Distinguish clea rl y b etwe en aCCliracy and pr ec ision.

S. State an d explain the cha rac teristics o f precision.

9. Exp laill t i le terms relative erro r a n d re lative perc entage error.

10 . What is scale s pan of an instrument?

11 . Define a dynamic response o f an instrument.

12 . D efin t' the following terms,i) Sp ee d o f r es po ns e ii) Lag ii i) F id eli ty ivY Dyna mic error .

13. Define and e xplain the types of errors possible in an instrument.

14 . Define limiting errors. Derive the e xpression for relative limiting error.

1 5 . A mov ing coil vo ltm ete r h as a u nifo rm s ca le with 100 divisions, the full scale reading is 200 V

an d 1/10 of sca le d ivisio n ca n be estim ated w ith a fa ir deg ree of certainity . Determine the

reso lution of the in strument i n volt . [Ans. : 0.2 V]

16. A digital vol tmeter has a read out range fr om 0 -9 999 counts. Determine the resolution of the

instrum ent in vo lt w hen the ful l scale reading is 9.999 \ I. [Ans. : 1 mV]

17. A t ru e value o f volta ge acro ss res ister is 50 V. T he i nstru men t r eads 49 V. Calculate

i) ab solute error ii) percentage error iii) percentage aCCliracy

[Ans. : 1 V, 2%, 98%]

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Voltmeters and Multimeters

The measurement of a given quantity is the result of comparison between the quantity

to be measured and a definite standard. The instruments which are used for such

measurements are called measuring instruments. The three basic quantities in the electricnlmC,lsurement are current, voltage and power. The measurement of these quantities is

important as it is used for obtaining measurement of some other quantity or used to test

the performance of some electronic circuits or components etc.

The necessary requirements for any measuring instruments are:

1) With the introduction of the instrument in the circuit, the circuit conditions

should not be altered. Thus the quantity to be measured should not get affected

due to the instrument used.

2) The power consumed by the instruments for their operation should be as small as

possible.

The instrument which measures the current flowing in the circuit is called ammeter

while the instrument which measures the voltage across any two points of a circuit is

c,ll1ed voltmeter. But there is no fundamental diff erence in the operating principle of

analog \'oltmeter and ammeter. The action of almost all the analog ammeters and

\oltmeters depends on the deflecting torque produced by an electric current. In ammeters

such c 1 torque is proportional to the current to be measured. In voltmeters this torque is

decided by a current which is proportional to the voltage to be measured. Thus all the

analog ammeters and voltmeters are basically current measuring devices.

A basic d.c. meter uses a motoring principle for its operation. It stntes that any current

carrying coil placed in a magnetic field experiences a force, which is proportional to the

magnitude of current passing through the coil. This movement of coil is called D'Arsonval

movement and basic meter is called D'Arsonval galvanometer. Adding various other

elements to the basic meter, various practical instruments can be obtained. These

instruments are classified as,

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a) Using shunt resistance, d.c. current can be measured. The instrument is d .c .

microammeter, milliammeter or ammeter.

b) Using series resistance called multiplier, d.c. voltage can be measured. The

instrument is d.c. millivoltmeter, voltmeter or kilovoltmeter.

c) Using a battery and resistive network, resistance can be measured. The instrument

is ohmmeter.

a) Using a rectifier, a.c. voltages can be measured, at power and audio frequencies.

The instrument is a.c. voltmeter.

b) Using a thermocouple type meter radio frequency (RF) voltage or current can be

measured.

c) Using a thermistor in a resistive bridge network, expanded scale for power line

voltage can be obtained.

The basic d.c. voltmeter is nothing but a

permanent magnet moving coil (PMMC)

0' Arsonval galvanometer. The resistance is

required to be connected in series with the

basic meter to use it as a voltmeter. This

series resistance is called a multiplier. The

main function of the multiplier is to limit the

current through the basic meter so that the

meter current does not exceed the full scale

deflection value. The voltmeter measures the

voltage across the two points of a circuit or a

voltage across a circuit component. The basic d.c. voltmeter is shown in the Fig. 2.1.

The voltmeter must be connected across the two points or a component, to measure

the potential difference, with the proper polarity.

The multiplier resistance can be calculated as :

Rm

Basicmeter

Let Rm internal resistance of coil i.e. meter

R s = series multiplier resistance

1m full scale deflection current

V = full range voltage to be measured

From Fig. 2.1, V = I In (R il l + R s)

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~v

L_'_-IIll_-

R

_

Ill

_

The multiplying factor for multiplier is the ratio of full range voltage to be I11casurC'd

and the drop across the basic cleter.

1 I" Vmu tlP VIng J,Ktor = -" - ~

IIll(RIll+R,)

'111 Rill

Thus to increase thE r

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(20 mA) Rm

10 n

20x 10 -' x 10200-20x 10 -'

Thi: is the required shunt.

ii) For llsing it as a v oltmeter,

500 V

~-R1

111

m

500 -1020x ]0--'

The range of the basic d.c. voltmeter can be extended by using number of multipliers

clnd a selector switch. Such a meter is called multirange voltmeter and is shown in

the Fig. 2.2.

Fig. 2.2 Multirange voltmeter

The R:, R2, R) and R~ are the four series multipliers. When connected in series with

the meter, they can give four diff erent voltage ranges as V], V2,V-, and V~. The "elector

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__ R_ :! R_T_-_(R_,,_n_+_R_3_+_R_~)__ JIn position V1, the multiplier is R1 + R:! + R3 + R~

VIRT =

Jm

R1 + R2 + R \ + R~ = F'I - Rill

RT - ( Rill + R2 + R3 + R~) JUsing the equations (1), (2), (3) and (4) multipliers can he dcsigncci. Tlw ,ld \ ,mtage of

this t

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v ,.-:- - Rm.till

_5_0 _ 50 - 49502x1O 1

_1_00 _ 50 - 4950 - 200002xlO 1

For position Vj = 500 V, multiplier is ( R1 + R2 + R, + R~ ).

VI(R1 + R2 + R, + R~) 1:- Rm

500 _ 50 - 4950 - 25000 - 200002xl0 3

R1 = 200 kQ

Thus Rj, R2, R, and R~ forms a series string of multipliers.

Tn a multirange voltmeter, the ratio of the total resistance R r to the vollag(' r,1ng:'

remains same.((rhis ratio is nothing but the reciprocal of the full scale deflection current ,)f

the meter)i.e. 1/101. This value is called sensitivity of the voltmeter.

Thus the sensitivity of the voltmeter is defined as,

S = I

Full scale deflection current

1S = r:D./V or kO .!V

Key Point: The sensiti()it y range is specified 0/1 the nleter dial and it ii7dicntes the res iston ce

of the 11Ieterfor n one volt range.

The internal resistance of the voltmeter is not the same in each of its ranges. The

higher is the range of the voltmeter, greater is its internal resistance. Internal resistance of

a voltmeter can be obtained from its sensitivity as,

Internal resistance of voltmeter = Maximum voltage (range) x Sensitivity in D/V

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The ~~'nsitivity i~ useful in calculating the resistance of a multiplier in d.c voltmeter.

Con-.;ider the prClcticcll multirange voltmeter circuit shown in the Fig. 2.5.

R1 R2 R3 R4

V2+

V1 t RmV4S BasIc

meter

Fig. 2.5

sensitivity rating in 0.jV

where VI, V2, V ., cllld V-I (lre the required voltage ranges.

Key Point: This lIlethod is called the sensitivity method of cnlClllating thl' lIlultiplier

/"l'sistallcl's.

1

2x10-3

= 500 OjV

VI 500 V, V2 = 1 0 0 V, VI = 50 V, V-I=10 V

R-l S V-l - Rm = 500 x 1 0 - 5 0

4.951

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111* Example 2.5: Calcula te the va lue of the mul tiplier re sistanc e 011 the 5UU V ra nge of 1 1 d.c.

utiltmeta, that us es 5U IlA meta movement with an in ternal resistance of 200 D.

s = r : , -50X~0 6 =20000 D/V

11I* Example 2.6 : The meter A has. a ral lgc of 0 - 100 V a nd m u ltip lie r r es istance of 25 ill.

TllC met('r B h a s a I"IIn ge 0 - 1000 V a nd a m u lt ipli('r re sistance of 150 kD.. Both metas /1I11'C

[Ia sic m ete r r es ista nc e o f 1 l\fl. Whi ch meter is 111 0resensitive?

Solution : ~or meter A, Rs = 25 ill, Rm= 1 ill, V = 100 V

Now R, SV- Rm

25 x 1() 3 S x 100 - lxlO 3

S2600.jV

For meter B, R, 150 ill, Rm= 1 ko., V = 1000 V

R, SV- Rm

150 x 103 S x 1000 - 1 x 103

S 151 D/V

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While selecting a m eter for a particular measurement, the sensitivity rating IS very

important. A low sensitive meter may give the accurate reading in low resistance circuit

but will produce totally inaccurate reading in high resistance circuit.

The voltmeter is always connected across the two points between which the potential

difference is to be measured. If it is connected across a low resistance then as voltmeter

resistance is high, most of the current will pass through a low resistance and will produce

the voltage drop which will be nothing but the true reading. But if the voltmeter is

connected tcr s arf used all the 150 V

+ 1

250 V

j

2 50 2 "-(20 + 25) x .J

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Req R2 I I Rv

25x 75

(25 + 75)

Hence the voltage across Rt.q is,

V = Req x 250(Req + R1)

Thus first voltmeter will read 120.96 V.

Case ii) S = 10,000 {1IV

The voltmeter resistance will be

Rv S V

1000,0 x 150

1.5 1 \.1 0

Req R2 I I Rv

25x1.5xl06 xl03

(25xl03 + 1.5xl06

)

Hence the voltage across Req is,

Req 250 24.59 ~50V = (Req + R1) x = (24.59 + 20) xL

Thus the second voltmeter reads more accurately.Key Point: Thus the high sensitivit y voltm eter gives more accurate read ing, though the

voltage range for both the meters is same.

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4.167 -4.132

4.167 x 100

0.84%

e) The percentage accuracy can be obtained as :

99.16%

Thus voltmeter 2 is 99.16% accurate while voltmeter 1 is 85.7% accurate.

2.5.2 Precautions to be taken while using a Voltmeter

The following precautions must be taken while using a voltmeter:1) The voltmeter resistance is very high and it should always be connected across the

circuit or component whose voltage is to be measured.

2) The polarities must be observed correctly. The wrong polarities deflect the pointer

in the opposite direction against the mechanical stop and this may damage the

pointer.

3) While using the multirange voltmeter, first use the highest range and then decrease

the voltage range until the sufficient deflection is obtained.

4) Take care of the loading effect. The effect can be minimised by using high

sensitivity voltmeters.

2.5.3 Requirements of a Multiplier

1) Their resistance should not change with time.

2) The change in their resistance with temperature should be small.

3) They should be non-inductively wound f or a.c. meters.

Commonly used resistive materials for construction of multiplier are manganin and

constantan.

The PMMC movement used in d.c. voltmeters can be effectively used in a.c.

voltmeters. The rectifier is used to convert a.c. voltage to be measured, to d.c. This d.c., if

required is amplified and then given to the PMMC movement. The PMMC movement

gives the deflection proportional to the quantity to be measured.

lt is important to study some basic definitions related to the a.c. quantities, before

studying the operation of the a.c. voltmeters. The a.c. meters are usually calibrated to read

Lm.S. value of an alternating quantity to be measured.

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The r.m.s. value of an alternating quantity is given by that steady current (d.c.) which

when flowing through a given circuit for a given time produces the same amount of heat

as produced by the alternating current which when flowing through the same circuit for

the same time. The r.m.s value is calculated by measuring the quantity at equal intervals

for one complete cycle. Then squaring each quantity, the average of squared v,llues is

obtained. The square root of this average value is the r.m.s. value. The r.m.s means

root-mean square i.e. squaring, finding the mean i.e. average and finally root.

If the waveform is continuous then instead of squaring and calculating mC,lll, the

integratioll is used. Mathematically the r.m.s. value of the continuous a.c. voltage having

time period T is given by,

1TT f V}, dt

o

The + term indicates the mean value or average value.

Most of the a.c. vo1tmeters are r.m.s. responding or average responding type, with

scale calibrated interms of the r.m.s. value of a sine wave.

The average value of an a.c. quantity is another important parameter. The average

value is defined as that value which is obtained by averaging all the instantaneous values

over a period of a half cycle. For the symmetrical a.c. quantity, the average value over a

complete cycle is zero as both positive and negative half cycles are exactly identical. Hence

average value is calculated over a half cycle. If the a.c. quantity is continuous then average

value can be expressed mathematically using an integration as,

T/2

Vav = ~ fVin dt

o

The interval T/2 indicates the average over half a cycle.

For purely sinusoidal quantity,

2= - Vm = 0.636

IT

As mentioned earlier, the average responding meter scale is also calibrated in terms of

r.m.s. values. To achieve such calibration, a pure sine wave ,,-ith ":'.m.s. value of 1 V is

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applied. Then the deflection of meter is adjusted to IV reading. For this, a particular factor

is required to be considered. This factor is called Form Factor.

The form factor is the ratio of r.m.s. value to the average value of an alternating

quantity.

r. m. s. value f f - - - - - - = orm actoraverage value

Key Point: For purely sinusoidal wave form theform factor is 1.11.

Thus while calibrating average responding meter interms of r.m.s. values the markings

are actually corrected by a factor of 1.11.

r. m. s. value

Kf

Some meter scales are calibrated in terms of peak values of the input. In such cases

another factor relating peak value and the r.m.s. value becomes important. This factor isGllled Peak Factor or Crest Factor.

The peak factor or crest factor is the ratio of peak (maximum) value to the r.m.s. value

of an alternating quantity.

K _ maximum valuep - r. m. s. value

Key Point: For purely sinusoidal a.c. quant ity the crest factor is 1.414.

The question is, why to use these factors to correct the readings by measuring averageand peak values, when the true r.m.s. voltmeter can give direct r.m.s reading. The reason

behind this is that the average and peak responding meters are less in cost and very

simple in construction as compared to true r.m.s. voltmeters.

A.C. voltmeters can be designed in two ways:

i) First rectifying the a.c. signal and then amplifying.

ii) First amplifying the a.c. signal and then rectifying.

2.6.1 First Rectifying and then Amplifying A.C. Signal

In this arrangement simple diode

rectifier circuit precedes the amplifier and

the meter. This is shown in the Fig. 2.8.

The a.c. input voltage if first rectified

using the diode D. This rectified signal is

then applied to the amplifier of gain A.

The amplified signal is then given to the

basic PMMC meter to obtain the

deflection.

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This

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diode 01 , the meter is shunted with a resistance Rsh' This ensures high current through

aiode and its linear behaviour.

When the a.c. input is applied, for the positive half cycle, the diode 01 conducts and

causes the meter deflection proportional to the average value of that half cycle.In the negative cycle, the diode O2 conducts and 01 is reverse biased. The current

through the meter is in opposite direction and hence meter movement is bypassed.

Thus due to diodes, the rectifying action produces pulsating d.c. and lile meter

indicates the average value of the input.

R Basicm meter

The a.c. voltmeter using half

wave rectifier is achieved byintroducing a diode in a basic d.c.

voltmeter. This is shown in the

Fig. 2.12.

The diode 0 conducts only

during positive half cycle. Let us

compare the sensitivities of d.c.

and a.c. voltmeters.

The sensitivity of d.c. voltmeter is,

I 5". = II Ifsd

Let I fsd be 1 mA, hence the sensitivity becomes 1 kO/volt. The series resistance Rs is

10 kO hence the 10 V d.c. input would cause exactly the full scale deflection, when

connected with proper polarity.

Let purely sinusoidal input of 1 0 V r.m.s. is applied.

Erms 10 V

Ep peak value = J 2 . Erms

Now the rectified d.c. is pulsating d.c. hence meter will deflect proportional to the

average value.

But the diode conducts only for half cycle and meter movement is bypassed for

another cycle. Hence it responds to half the average value of the a.c. input.

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= 8.99", 4 5 V2 .

Thus pointer will deflect for full scale if 10 V d.c. is applied and 4.5 V when 10 V

r.m.s. sinusoidal input is applied.

I Ede = 0.45 Erms I

Thus the vi:l1ue of series multiplier can be obtained for a.c. voltmeters as,

Rs

= E ] de :... Rm

de

OASE rms----- RmIde

The a.c. voltmeter using full wave rectifier is achieved by using bridge rectifier

consisting of four diodes, as shown in the Fig. 2.13.

Fig. 2.13 A.C. voltmeter using full wave rectifier

Let 10 V r.m.s. purely sinu?oidal input be applied.

0.636 Ep = 8.99 V

"" 9 V

Now this meter uses full wave rectifier and hence the average value of output over a

cycle is same as average of the input over a cycle i.e. 9 V.

Thus, the 10 V r.m.s. voltage is equal to 9 V d . c. for full scale def lection Thus the

jPointer will deflect to 90% of full scale.

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R, =Elk -

RrnIde

)1. Example 2.9 : All a.c. 'uoltmeter IIses half wave rectifier alld the basic I/ /e ter with filII

scale d{:flection current of 1 mA and tire meter resistallce of 200 D. Calclliate the 1IlIIItiplier

resistance for a 10 V 1'.111.5.rallge on the voltmeter.

Solution: The meter uses half wave rectifier and input is 10 V r.m.s.

Eav ~ (Eav over a cycle of input)

0.6 Ep = 8.99 '" 9 V

1"2 x 9 = 4.5 V

Eo e R-J-- III

~

0.45x JO _ 200

Ix 10'

)1. Example 2.10: All a.c. voltmeter IIses 11filII wave bridge rectifier and tire basic I//eter

ll'itlr filII scale deflection current of 2 I1IA I 1l1a tire meter resistance of 500 D. Calclliate the

1IlIIItiplier resistl1nce for 11 10 V 1'./11.5. rallge all the voltmeter.

Solution : The meter uses f ull wave rectifier,

0.9x 10 _ 500 = 4000 D2x 10 3

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As the name indicates, this type of meter responds to the peak value of the