Chapter 0004

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Measurement of ResistancelInductance and Capacitance

The bridges are used for not only the measurement of resistances but also used for the

measurement of various component values like capacitance, inductance etc.

A bridge circuit in its simplest form consists of a network of four resistance arms

forming a closed circuit. A source of current is applied to two opposi te junctions. The

current detector is connected to other two junctions.

The bridge circuits use the comparison measurement methods and operate on

null-indication principle. The bridge circuit compares the value of an unknown

component with that of an accurately known standard component. Thus the accuracy

depends on the bridge components and not on the null indicator. Hence high degree of

accuracy can be obtained.

In a bridge circuit, when no current flows through the null detector which is generally

ga\y,mome\er, \'he bl"\dge \s said to be balanced. 'The relahom,hlp between the component

values of the four arms of the bridge at the balancing is called balancing condition or

balancing equation. This equation gives us the value of the unknown component.

7.1.1 Advantages of Bridge Circuit

The various advantages of the bridge circuit are,fls'1'16\...~o.""

1) The balance equation is independent of the magnitude of the input voltage or its

source impedance. These quantities do not appear in the balance equation

expression.

2) The measurement accuracy is high as the measurement is done by comparing the

unknown value with the standard value.

3) The accuracy is independent of the characteristics of a null detector and is

dependent on the component values.

4) The balance equation is independent of the sensitivity of the null detector, the

impedance of the detector or any impedance shunting the detector.

S) The balance condition remains unchanged if the source and detector are

in terchanged.

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Measurement of Resistance,

Inductance and Capacitance

6) The bridge circuit can be used in the control circuits. When used in such control

applications, one arm of the bridge contains a resistive element that is sensitive to

the physical parameter like pressure, temperature etc. which is to be controlled.

The two types of bridges are,

1) D.C. bridges and 2) A.c. bridges

The d.c. bridges are used to measure the resistances while the a.c. bridges are used to

measure the impedances consisting capacitances and inductances. The d.c. bridges use the

d.c. voltage as the excitation voltage while the a.c. bridges use the alternating voltage as

the excitation voltage.

The two types of d.c. bridges are,

1. Wlwatstone bridge 2. Kelvin bridge

The vilrious types of a.c. bridges. are,

1. Capacitilnce comparison bridge

3. Maxwell's bridge

5. Anderson bridge

7. Wien bridge

let us discuss in detail, the various types of bridges.

2. Inductance comparison bridge

4. Hay's bridge

6. Schering bridge

The bridge consists of four resistive arms together with a source of e.m.f. and a null

detector. The galvanometer is used as a null detector.

The Fig. 7.1 shows the basic Wheatstone bridge circuit.

Ratio

arms

Unknown

resistanceStandard

resistance

+ 1 11 -

E

Fig. 7.1 Wheatstone bridge

The

consistinl

the unkn

galvanorr

Wher

show an

indicatiol

To hi

potential.

Thus

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The arms consisting the resistances R] and R2 are called ratio arms. The arm

consisting the standard k nown resistance R3 is called standard arm. The resistance R4 is

the unknown resistance to be measured. The battery is connected between A and C while

galvanometer is connected between Band D.

When the bridge is balanced, the galvanometer carries zero current and it does not

show a!'y deflection. Thus bridge works on the principle of null deflection or null

indication.

To have zero current through galvanometer, the points Band 0 must be at the same

potential. Thus potential across arm AB must be same as the potential ,Kross clrm AD.

Considering the battery path under balanced condition,

EI] :=:J 3 :=:---

R J +R1

Using (3) and (4) in (1),

E E----xR] ---xR4R]+R2 R3+R4

R] (R 3 +R4)

R] R3 +R] Ri

R4(R]+R2)

R.] R4 +R 2 R4

This is required balance condition of Wheatstone bridge.

The following points can be observed.

1. It depends on the ratio of R] and R2 hence these arms are called ratio arms.

2. As it works on null indication, the results are not dependent on the calibration and

characteristics of galvanometer.

3. The standard resistance R3 can be varied to obtain the required balance.

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Req B

~

WI/v

T I ' gVTH G Rg

D

0.444

4.888 x 10 3 + 300

The Fig. 7.10 is the basic circuit of the

Kelvin bridge.

The resistance Rv represents the

resistance of the connecting leads from R .,

to R,. The resistance Rx is the unknown resistance to be measured.

+ I I -E

In the Wheatstone bridge, the bridge

contact and lead resistance causes significant

error, while measuring low resistances. Thus

for measuring the values of resistance below

1 - n , the modified form of Wh~tstonebridge is used, known as Kelvin bridge. The

consideration of the effect of contact and

lead resistances is the basic aim of the

Kelvin bridge.

The galvanometer can be connected to either terminal a, b o r t erminal c. When it is

connected to a, the lead resistance Ry gets added to Rx hence the value measured by the

bridge, indicates much higher value of Rx .

If the galvanometer is connected to terminal c, then Ry gets added to R3. This results

in the measurement of Rx much lower than the actual value.

The point b is i n between the points a and c, in such a way that the ratio of the

resistance from c to b and that from a to b is equal to the ratio of R] and R2.

R cb _ R ]R

ab- - R 2

But R3 and Rx now are changed to R) + Rab and Rx + Rcb respectively due to lead

resistance.

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Electronic Instrumentation 7 -13

R 1 ( R 3 + R ~b) R z( Rx + R eb)

R x + R cb) = ~ (R, + Rab)R2

-

Now we have Reo "R 1

, R"" Rz

R ebl R

j 1-+ =-+R al, R2

R eb +R ab R j ---R2

R,'b Rz

But R cb +Rab =R"

Substituting in (5) we get,

R: Rj +R2

R ab R2

R ab =RzR y

R] +R 2

Now R cb + Rab = R v



r R.,!R '1- - Iv l R]+ RzJ

RjRy

R] +R2

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Thus equation (10) represents standard bridge balance equation for the Wheatstone

bridge. Thus the effect of the connecting lead resistance is completely eliminated by

connecting the galvanometer to an intermediate position 'b'.

This principle forms the basis of the construction of Kelvin's Double Bridge which is

popularly called Kelvin Bridge.

+ I I -E

Fig. 7.11 Kelvin's double bridge

This bridge consists of another set of

ratio arms hence called double bridge. The

Fig. 7.11 shows the circuit diagra-ffi of

Kelvin's Double Bridge.

The second set of ratio arms is t he

resistances 'a' and 'b'. With the help of these

resistances the galvanometer is connected topoint '3'. The galvanometer gives nu ll

indication when the potential of the

terminal '3' is same as the potential of the

terminal '4'.

The ratio of the resistances a and b is same as the ratio of R] and R2.

a R1

b R2

'.Consider the path from 5-1-2-6 back to 5 through the battery E. The resistance betweenthe terminals 1-2 is the parallel combination of Ry and (a + b).

I x [R 3 +R yll(a+b)+R x J

[

(a + b) Ry ]IR 3+R x + bR

a + + v

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For Esn, consider the path from the terminal 5 to 2 as shown in the Fig. 712.

Galvanometer carries zero current

Now from the Fig. 7.12 we can write,

I X [RY (3+b)lV12 = - - - -

Ry+3+b...J

b

- - - . V i?a+ b -

_b_ .I r_R_y_(_a~+~b_)l J

a+b L...Ry+a+b

I R3 + VB

= IR +I~_[Ry(a+b)]3 a+b Ry +a+b

I R2 [ (a+ b)Rv J

l

--- R3+Rx +a+b+R"yR , + R 2

, - b { R y (a+ b ) } lI R, +-- -------

..) a+b a+b+Ry

L . J

R] +R2 ' Ir b r Ry (a+ b ) } ]---- R,+--~----

R2

l-a + b la + b+ Ry

R]R3 bRy R]bRy (3+ b) R]

R2-+Ry+~+b+R2(Ry+a+b) - CRy -;-a+b)

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R]R3 bR] Ry--+------

R2 R2 (Ry + a + b)

Rx = R]R3 + bRy [RR]2-~b]

R2 (Ry + a + b)

This is the standard equation of the bridge balance.The resistances a, band Ry are not

present in this equation. Thus the effect of lead and contact resistances is completely

eliminated.

Key Point: The importan t condi tion f or this bridge b ala nce co nd ition is that the ratio ~fthe res istances of ratio ar ms must be sam e as tile ratio of tile 1'esista 11c es of the second ratio

l11'ms.

In a typical Kelvin's double bridge, the range of a resistance covered is 10 to 10 pQ

with an accuracy of 005 % to 0.2 %.

~ractical Kelvin's Double Bridge

, The Fig. 7.13 shows a commercial Kelvin's double bridge, capable of measuring the

resistances of very low range from 10 nto 0.00001 O.

The resistance R 3 is replaced by the standard resistance consisting on nine steps of0001 n each, plus a calibrated manganin bar of 0.0011'"0 with a sliding contact. The

required resistance can be selected by the switch S 3. The total resistance of the R " arm

amounts to 0.0101 nand is variable in steps of 0.001 nplus fractions of 0.0011 nby thesliding contact.

When both the contacts are switched to select" the proper value of standard resistance,

the voltage drop between the ratio arm connection points is changed but the total

resistance around the battery circuit is unchanged.

With this arrangement any contact resistance can be placed in series with the relatively

high resistance values of the ratio arms. Due to this, the effect of contact resistance

becomes negligibly small.

The ratio of R] and R2 is selected in such a way that the larger part of the variable

standard resistance i.s used and hence Rx is determined to the largest possible number of

significant figures. This increases the measurement accuracy.

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ParameterRv sensitive

resistanceError

signal

!

7."'/ A.C. Bridges

An a.c. bridge in its basic form consists of four arms, a source of excitation and a

balance detector. Each arm consists of an impedance. The source is an a.c. supply which

supplies a.c. voltage at the required frequency. For high frequencies, the electronic

oscillators are used as the source. The balance detectors commonly used for a.c. bridge

are head phones, tunable amplifier circuits or vibration galvanometers. The headphones are

used as detectors at the frequencies of 250 Hz to 3 to 4 kHz. While working with single

frequency a tuned detector is the most sensitive detector. The vibration galvanometers are

useful for low audio frequency range from 5 Hz to 1000 Hz but are commonly used below

200 Hz. Tunable amplifier detectors are used for frequency range of 10 Hz to 100 Hz.

The simple a.c. bridge is

Head phone as null the outcome of the Wheatstonedetector

bridge. The impedances at

audio and radio frequency

range can be easily determined

by such simple a.c. Wheatstone

bridge. It is shown in the

Fig. 7.20.

This is similar to doc.

Wheatstone bridge. The bridge

arms are impedances. The

bridge is excited by a.c. supply and pair of headphones is used as a null detector. The null

response is obtained by varying one of the bridge arms.

a.c.supply

For bridge measurements at very low frequencies, the power line itself may act as asource of supply to the bridge circuit. For bridge measurements at higher frequencies

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Solution : From the given bridge,

Zj = 50 L 40Q, Zz = 100 L - 90Q, Z3 = 15L 45 Q, Z4 = 30 L 30Q

The bridge balance equation is,

Z] Z4 = Z2Z3

Equating magnitudes, I Z] Z41 = I Z 2 Z 31

IZ1Z41 = SOx 30 = 1500 and IZ2Z31 = 100 xIS = 1500

82 + 83 ... Angle condition

40 + 30 = 70 and 82 + 83 = - 90 + 45 = - 45

Thus angle condition is not satisfied.

Hence the bridge is not under balanced condition.

~ Capacitance Comparison Bridge

a.cSupply rv

f Hz

In the capacitance comparison bridge

the ratio arms are resistive in nature. The

impedance Z 3 consists of the known

standard capacitor C3 in series with the

resistance R3. The resistance R3 is

variable, used to balance the bridge. The

impedance Z 4 consists of the unknown

capacitor Cx and its small leakage

resistance Rx .

The unknown capacitor Cx is

Fig. 7.22 Capacitance comparison bridge compared with the standard capacitor. By

using the balance equation, the capacitor

and its leakage resistance value is obtained. The bridge is shown in the Fig. 7.22.

Here Z] R ] + j O Q,

Z2 R 2 + j O Q,

Z3 R 3-jXC3 = R 3 -j( ~) QWC3

Z4 R x - j Xcx

R x _j ( _1 ) .0.wCx

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R] Rz

wCx wC3

I C. _ C3R1-~By using equations (1) and (2) the unknown capacitor and its leakage resistance can be

determined. By varying R] and R3 simultaneously, true balance can be obtained.

m . Example 7.6 : .A capacitance comparison bridge is used to measure the capacitiveimpedance at a frequecy of 3 kHz. The bridge constants at bridge balance are,

C3 10 ~F

R) 1.2 kn

R2 100 kn

R3 = 120 kn

Find the equivalent series circuit of the unknown impedance.

Solution : From the bridge balance equations,

Rz R3 100x103x120x103Rx = =

R1 1.2x103

= 10 Mn

while CxR1 C3 1.2x10

3 x10x10-6

= ~- 100x 103

= 0.12 ~F

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By using equations (1) and (2) , unknown elements can be determined.

In this bridge, R2 is selected as inductive balance control and R3 as resistance balance

control. The balance is obtained by alternately varying L3 or R,.

))I. Example 7.7: An inductance co mp arison brid ge is used to measure the ind uctive

impedance at a frequency of 1.5 kHz. The bridge constants at bridge balance are,

L, = 8 mH, Rj = 1 k 0 , R2 = 25 k 0, R3 = 50 k 0

Find the equivalent series circuit of unknown impedance.

Solution : From bridge balance equation of inductance comparison bridge,

RxR2 R3 25 x 10

3 x 50 x 10 3= 1.25 Mn=

R1 1x10

3

R2 L325 x 1 a 3 x 8 x 10 -3

= 200 mHand L x -R1 1x10 3

200 mH 1.25Mn

4 Fig. 7.25~jVMaxwell's Bridge

Maxwell's bridge can be used to measure inductance by comparison either with a

variable standard self inductance or with a standard variable capacitance. These two

measurements can be done by using the Maxwell's bridge in two different forms.

7.10.1 Maxwell's Inductance Bridge

Using this bridge, we can measure inductance by comparing it with a standard

variable self inductance arranged in bridge circuit as shown in Fig. 7.26 (a).

Consider Maxwell's inductance bridge as shown in the Fig 7.26 (a). Two branches

consist of non-inductive resistances R1 and Rz. One of the arms consists variable

inductance with series resistance r. The remaining am1 consists unknown inductance L x .

At balance, we get condition as

R1

[(R3 + r)+ j c oL3]

Rz[(R 3+r)+jcoL31

Rz

(R 3 + r)+ jcoRz

L3

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a.c.supply rvV

f Hz

" " ; ~ ' "~ ' : < ~ - - - - - - - - - - - - - - - - - -V2

>~:: :'S~R, , . e # / j

V3 , , : :? v~}, ',

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The resistances are expressed in ohms, the inductances in henries and capacitance in

farads.

The quality factor of the coil is given by,

Q = w L x w R 2 R 3 C 1

R x ( R ~ ~ 3 )

I Q = W R1 C1 I

The advantages of using standard known capacitor for measurement are:

1) The capacitors are less expensive than stable and accurate standard inductors.

2) The capacitors are almost lossless.

3) External fields have less effect on a capacitor. The standard inductor requires well

shielding in order to eliminate the effect of stray magnetic fields.

- 1 ) The standard inductor will not present its rated value of inductance unless current

flow through it is precisely adjusted.

5) The capacitors are smaller in size.

This bridge is also called Maxwell Wien bridge.

7.10.3 Advantages of Maxwell Bridge

The advantages of the Maxwell bridge are:

J4 The balance equation is independent of losses associated with inductance.

f? The balance equation is independent of frequency of measurement.

WThe scale of the resistance can be calibrated to read the inductance directly.

-1 ) The scale of R1 can be calibrated to read the Q value directly.

5). When the bridge is balanced, the only component in series with coil under test is

resistance R2 If R2 is selected such that it can carry high current, then heavy

current carrying capacity coils can be tested using this bridge.

7.10.4 Limitations of Maxwell Bridge

The limitations of the Maxwell bridge are:

1) It cannot be used for the measurement of high Q values) Its use is limited to the

measurement of low Q values from 1 to 10k-This can be proved from phase angle

balance condition which says that sum of the angles of one pai.r of opposite arms

must be equal.

Chapter 0004

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