Chapter 0004
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Transcript of 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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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
Measurement of Resistance,
Inductance and Capacitance
r R.,!R '1- - Iv l R]+ RzJ
RjRy
R] +R2
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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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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Measurement of Resistance,
Inductance and Capacitance
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.