Electronic Instrumentation 15EC35

MODULE 4
BRIDGES
Introduction:
 Bridges is a circuit which is used for measuring various components like R, C and L
 Bridge as a simple circuit consists of having 4 resistance arms in a closed loop, with
dc current source applied to 2 opposite junction and current detector connected to other
2 junction as shown in Fig. 4.1.
 In this the unknown component is measured in comparison with known component
called as standard.


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This method of measurement is very accurate and the accuracy of measurement is
directly proportional to the bridge component.

There are 2 types of bridges

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 ac bridge – impedances consisting of C and L
 dc bridges – measure resistance
The dc bridge used for measuring resistance is called Wheatstone’s bridge.
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Wheatstone’s Bridge:
 It is the most accurate method for measuring resistance and a common method used
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in laboratory.
 The circuit is shown in Fig 4.1.
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 It has an emf source E and switch Sw connected between points A and B.

 A sensitive current indicating meter is connected to point C and D. Meter used is a
zero center scale, when at rest it is mid scale at 0 current. Current in one direction
causes pointer to deflect in one direction and for current in the opposite direction
causes the pointer to deflect in opposite direction. When no current flowing in the
circuit, the pointer rests at ‘0’.
 When Sw is closed current flows and divides into 2 arms at point A as I1 and I2.

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

 Bridge is balanced when current through G is ‘0’ i.e potential difference at C and D
should be equal.
i.e 𝐼1𝑅1 = 𝐼2𝑅2 ------------ (1)
For galvanometer current to be zero, I1=I3 and I2=I4
Thus under balanced condition,
𝐸
𝐼1 = 𝐼3 = (𝑎)
𝑅1 + 𝑅3
And
𝐸
𝐼2 = 𝐼4 = (𝑏)
𝑅2 + 𝑅4
Using (a) and (b) in equation (1), we get
𝐸 ∗ 𝑅1 𝐸 ∗ 𝑅2

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=
𝑅1 + 𝑅3 𝑅2 + 𝑅4
Simplifying the above equation we get,
𝑅1(𝑅2 + 𝑅4) = 𝑅2(𝑅1 + 𝑅3)

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𝑅1𝑅2 + 𝑅1𝑅4 = 𝑅1𝑅2 + 𝑅2𝑅3
𝑅1𝑅4 = 𝑅2𝑅3
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𝑅2 𝑅3
Now 𝑅4 =
𝑅1

This is the equation for bridge to be balance.

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For balancing one of the resistance will be made adjustable and if R4 is the unknown
resistance then
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𝑅2 𝑅3
𝑅𝑥 =
𝑅1
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Sensitivity of Wheatstone’s bridge:

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 When there is unbalance in the bridge, there is deflection in the pointer of

galvanometer (G) which depends on the sensitivity of the galvanometer.

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If the G is more sensitive then, the deflection is more for the same amount of current.
Thus sensitivity is considered as deflection/unit current. i.e S = D/I , D = deflection
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and I = current in µA
 Sensitivity can be expressed in linear or angular with the units as S = mm/µA
(Linear) and S = degree/µA or S = radian/µA (Angular)
 Thus total deflection D = S * I

Unbalanced Wheatstone’s bridge:

This is the analysis of Wheatstone’s bridge under unbalanced condition and this determines
the amount of current flowing in the G.
The circuit analysis can be done using any general circuit analysis, considering “Thevenin’s
Theorem” will determine the current through G.

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

Since the interest is to find the current through G under unbalanced condition we need to
find the Thevenin’s equivalent circuit as seen by G
The first step is to remove G and find open circuit voltage between terminals a and b as
shown in fig 4.2

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Applying voltage divider at point ‘a’ and ‘b’, we get
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𝐸𝑎 =
𝑅1 + 𝑅3
𝑅4 𝐸
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𝐸𝑏 =
𝑅2 + 𝑅4
Voltage between a and b is the difference between Ea and Eb and this represents the
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Thevenin’s equivalent voltage,

𝑅3 𝐸 𝑅4 𝐸
𝐸𝑡ℎ = 𝐸𝑎 − 𝐸𝑏 = −
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𝑅1 + 𝑅3 𝑅2 + 𝑅4
𝑅3 𝐸 𝑅4 𝐸
Thus 𝐸𝑡ℎ = 𝐸 ( − )
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𝑅1+𝑅3 𝑅2+𝑅4

Thevenin’s equivalent resistance can be determined by replacing voltage source by its

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internal impedance or with a short and looking into a and b as shown in fig 4.3.,
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𝑅𝑡ℎ = (𝑅1 ‖ 𝑅3) + (𝑅2 ‖ 𝑅4)

𝑅1 𝑅3 𝑅2 𝑅4
Thus 𝑅𝑡ℎ = +
𝑅1+𝑅3 𝑅2+𝑅4

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

Thevenin’s equivalent circuit is shown in fig 4.4

If G is connected between a and b in the above circuit and its original circuit then both
experencies same deflection.

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The magnitude of current is limited by Rth and the resistance seen with G i.e Rg (internal
resistance of G)
Thus the deflection of current in galvanometer is given by

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𝐸𝑡ℎ
𝐼𝑔 =
𝑅𝑡ℎ + 𝑅𝑔
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Slightly unbalanced Wheatstone’s bridge:
If three of the four resistor in a bridge are equal to R and the fourth differs by 5% or less, we
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can develop an approximate but accurate expression for Thevenin’s equivalent voltage
and resistance as follows. The circuit is shown in Fig 4.5
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Voltage at point ‘a’ is given by

𝐸𝑅 𝐸𝑅 𝐸
𝐸𝑎 = = =
𝑅+𝑅 2𝑅 2
Voltage at point ‘b’ is given by
𝐸 (𝑅 + 𝛥𝑟) 𝐸 (𝑅 + 𝛥𝑟)
𝐸𝑏 = =
𝑅 + (𝑅 + 𝛥𝑟) 2𝑅 + 𝛥𝑟
Thevenin’s equivalent voltage is given by,

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

𝐸𝑡ℎ = 𝐸𝑎 − 𝐸𝑏 = 𝐸𝑏 − 𝐸𝑎
𝐸 (𝑅 + 𝛥𝑟) 𝐸
𝐸𝑡ℎ = −
2𝑅 + 𝛥𝑟 2
Simplifying this,
2𝑅 + 2𝛥𝑟 − 2𝑅 − 𝛥𝑟
𝐸𝑡ℎ = 𝐸 ( )
2(2𝑅 + 𝛥𝑟)

𝛥𝑟
𝐸𝑡ℎ = 𝐸 ( )
4𝑅 + 2𝛥𝑟
Now if 𝛥𝑟 is 5% of R or less, then 𝛥𝑟 can be neglected at the denominator without

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appreciable error. Thus Eth now is
𝛥𝑟
𝐸𝑡ℎ =
4𝑅

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The equivalent resistance can be calculated by replacing the voltage source with its internal
impedance,
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𝑅𝑅 𝑅 (𝑅 + 𝛥𝑟)
𝑅𝑡ℎ = +
𝑅 + 𝑅 𝑅 + (𝑅 + 𝛥𝑟)
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Simplifying the above equation,
𝑅 𝑅 (𝑅 + 𝛥𝑟)
𝑅𝑡ℎ = +
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2 2𝑅 + 𝛥𝑟
If 𝛥𝑟 is small compared to R, then it can be neglected
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𝑅 𝑅 𝑅)
𝑅𝑡ℎ =
+
2 2𝑅
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𝑅 𝑅 2𝑅
𝑅𝑡ℎ = + = =𝑅
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2 2 2
Thus the Thevenin’s equivalent circuit is shown in Fig 4.6
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The current through the G is given by,

𝐸𝑡ℎ
𝐼𝑔 =
𝑅𝑡ℎ + 𝑅𝑔
Applications of Wheatstone’s bridge:
 Wheatstone bridge is used to measure resistance in the range of 1Ω to low MΩ.
 Used to measure the dc resistance of various types of wire, either for the purpose of
quality control of the wire itself, or of some assembly in which it is used.
 To find the resistance of motor windings, transformers, solenoids, and relay coils.
 Wheatstone Bridge Circuit is also used extensively by telephone companies and others
to locate cable faults.
Dept. of ECE, SVIT 2017-18
Electronic Instrumentation 15EC35

Advantages of Wheatstone’s bridge:

 It operates on null deflection i.e., indication is independent on indicating instrument's
characteristics and this is reason it has high degrees of accuracy.
 The variation in the source does not alter the balance of bridge, hence the
corresponding errors are completely avoided.
 In Wheatstone bridge potential errors are canceled out including the bridge
excitation, and temperature errors.

Limitations of Wheatstone’s bridge:

 For low resistance measurement, the resistance of the leads and contacts becomes
significant and this may introduce error.

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 While measuring high resistance, the resistance presented by the bridge becomes so
large that the galvanometer will be insensitive to imbalance. Thus for high resistance
measurements in mega ohms, the Wheatstones bridge cannot be used.
 Another problem in Wheatstone Bridge Circuit is the change in resistance of the bridge
arms due to the heating effect of current through the resistance. The rise in temperature

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causes a change in the value of the resistance, and sometimes high current may cause
a permanent change in value.
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Kelvin’s bridge
 When the resistance to be measured is of the order of magnitude of bridge contact and
lead resistance, a modified form of Wheatstone’s bridge, the Kelvin's bridge is used.
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 Kelvin's bridge is used to measure values of resistance below 1 Ω. In low resistance
measurement, the resistance of the leads connecting the unknown resistance to the
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terminal of the bridge circuit may affect the measurement.

 Thus in Kelvin’s bridge, the effect of contact and lead resistance is important.
 Consider the circuit in Fig.4.7, where Ry represents the resistance of the connecting
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leads from R3 to Rx (unknown resistance). The galvanometer can be connected either

to point c or to point a.
 When it is connected to point a, the resistance Ry, of the connecting lead is added to
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the unknown resistance Rx, resulting in too high indication for Rx.
 When the connection is made to point c, R3, is added to the bridge arm R3 and
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resulting measurement of Rx is lower than the actual value.

 If the galvanometer is connected to point b, in between points c and a, in such a way
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that the ratio of the resistance from c to b and that from a to b equals the ratio of
resistances R1 and R2, then

𝑅𝑐𝑏 𝑅1
=
𝑅𝑎𝑏 𝑅2

The bridge balance equation is given by,

R1* R3 = R2 * Rx

But R3 is now R3 +Rab and Rx is now Rx + Rcb

Therefore R1 * (R3 +Rab) = R2 * (Rx + Rcb) -------------- (1)

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

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From the Fig 4.7,
Rab + Rcb = Ry ----------------- (A)

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and
𝑅𝑐𝑏 𝑅1
= − − − − − (𝐵)
𝑅𝑎𝑏 𝑅2
Adding 1 to both sides of equation (B), we get
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𝑅𝑐𝑏 𝑅1
+1 = +1
𝑅𝑎𝑏 𝑅2
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𝑅𝑐𝑏 + 𝑅𝑎𝑏 𝑅1 + 𝑅2
=
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𝑅𝑎𝑏 𝑅2
using equation (A) in the above equation, we get

𝑅𝑐𝑏 + 𝑅𝑎𝑏 𝑅𝑦 𝑅1 + 𝑅2
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= =
𝑅𝑎𝑏 𝑅𝑎𝑏 𝑅2
and Rab is now
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𝑅2 ∗ 𝑅𝑦
𝑅𝑎𝑏 =
𝑅1 + 𝑅2
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Rearranging equation (A) and using equation for Rab,

Rcb = Ry − Rab
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𝑅2 ∗ 𝑅𝑦
Rcb = Ry −
𝑅1 + 𝑅2
𝑅1𝑅𝑦 + 𝑅2𝑅𝑦 − 𝑅2𝑅𝑦
𝑅𝑐𝑏 =
𝑅1 + 𝑅2
𝑅1 𝑅𝑦
𝑅𝑐𝑏 =
𝑅1 + 𝑅2
Substituting Rab and Rcb in equation (1)
𝑅2 𝑅𝑦 𝑅1 𝑅𝑦
R1 * (R3 + ) = R2 * (Rx + )
𝑅1+𝑅2 𝑅1+𝑅2

Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

𝑅1 𝑅𝑦 𝑅1 𝑅2 𝑅𝑦
𝑅𝑥 + = (R3 + )
𝑅1 + 𝑅2 𝑅2 𝑅1 + 𝑅2

𝑅1 𝑅𝑦 𝑅1 𝑅3 𝑅2 𝑅𝑦 𝑅1
𝑅𝑥 + = + ∗
𝑅1 + 𝑅2 𝑅2 𝑅1 + 𝑅2 𝑅2
𝑅1 𝑅𝑦 𝑅1 𝑅3 𝑅1 𝑅𝑦
𝑅𝑥 + = +
𝑅1 + 𝑅2 𝑅2 𝑅1 + 𝑅2
Upon simplification, we get

𝑅1 𝑅3
𝑅𝑥 =
𝑅2
 The above equation is the normal Wheatstone’s bridge under balanced condition.
 Also the effect of lead resistance connecting from a to c is eliminated by connecting

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galvanometer to intermediate position ‘b’.
 This is the principle of constructing Kelvin’s double bridge also known as Kelvin’s bridge. It
is called double bridge as it incorporates 2nd set of resistance ratio arms.
 The schematic of Kelvin’s double bridge is shown in Fig 4.8
 In this 2nd set of arms a and b connect galvanometer to point c

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 The galvanometer gives null indication when potential at k and c are equal i.e 𝐸𝑙𝑘 = 𝐸𝑙𝑚𝑐
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Now Elk is given by

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𝑅2 𝐸
𝐸𝑙𝑘 =
𝑅1 + 𝑅2
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and E is given by
E=I*R
𝑅2
𝐸𝑙𝑘 = ∗𝐼∗𝑅
𝑅1 + 𝑅2

R is to determined considering path l-m-n-o-l, in this path at point m – n, the resistance is

Ry ‖ (a+b). Thus total resistance now becomes as
𝑅𝑦 (𝑎 + 𝑏)
𝑅 = 𝑅3 + 𝑅𝑥 +
𝑎 + 𝑏 + 𝑅𝑦
Now Elk is given by

𝑅2 𝑅𝑦 (𝑎 + 𝑏)
𝐸𝑙𝑘 = ∗ 𝐼 ∗ [𝑅3 + 𝑅𝑥 + ]
𝑅1 + 𝑅2 𝑎 + 𝑏 + 𝑅𝑦
Dept. of ECE, SVIT 2017-18
Electronic Instrumentation 15EC35

Similarly Elmc is given by, Elmc = Elm +Emc

Elm = I R3

𝑏 𝐸𝑚𝑛
𝐸𝑚𝑐 = using voltage divider rule
𝑎+𝑏

𝑅𝑦 (𝑎+𝑏)
𝐸𝑚𝑛 = (𝑎 + 𝑏)‖ 𝑅𝑦 ∗ 𝐼 = *I
𝑎+𝑏+𝑅𝑦
Thus now
𝑏 𝑅𝑦 (𝑎 + 𝑏)
𝐸𝑚𝑐 = ∗ 𝐼∗
𝑎+𝑏 𝑎 + 𝑏 + 𝑅𝑦
Therefor
𝑏 𝑅𝑦 (𝑎 + 𝑏)

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𝐸𝑙𝑚𝑐 = 𝐼 𝑅3 + ∗ 𝐼∗
𝑎+𝑏 𝑎 + 𝑏 + 𝑅𝑦

𝑏 𝑅𝑦 (𝑎 + 𝑏)
𝐸𝑙𝑚𝑐 = 𝐼 [𝑅3 + ∗ ]
𝑎 + 𝑏 𝑎 + 𝑏 + 𝑅𝑦

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But under balanced condition,
𝐸𝑙𝑘 = 𝐸𝑙𝑚𝑐
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Dept. of ECE, SVIT 2017-18

Electronic Instrumentation 15EC35

 The above is the equation for Kelvin’s bridge.

 From the above equation, Ry i.e., resistance of the connecting lead has no effect on
the measurement provided the resistance ratio of arms are equal.

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 This bridge can measure resistance in the range of 1Ω - 10μΩ with accuracy of ±0.05%
to ±0.02%.

AC Bridges:
 The ac bridges are similar to dc bridge except that the bridge arms have impedances

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and the bridge is excited by ac source rather than dc source.
 Impedances at audio frequency and radio frequency can be determined by means of
ac bridges.
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Dept. of ECE, SVIT 2017-18

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