Electrical Measurements

AC Bridges
In the alternating current bridge are the best and most used method for the precise measurement of
self and mutual inductance and capacitances. The supply is from an alternating current source. If the
supply is at normal commercial frequencies then a vibration galvanometer is used, whereas at high
frequencies a headphone is employed as detector. The alternating current bridge networks may be
classified in the following three classes:

i) Wheatstone impedance networks without mutual inductance.

ii) Wheatstone networks in which there are mutual inductances between appropriate pairs of
branches.
iii) Networks which are not of Wheatstone form and in which there are usually mutual
inductances.

Wheatstone Networks:

The difference from the dc Wheatstone bridge is that the four branches have, in general, impedance in
place of resistances as shown in figure.

Under the balance condition, if Z1 is the unknown impedance then

…………………………………(1)

In general, if R and X represent the resistance and reactance of the branches then

……………………….(2)

Now, to make the resistance A independent of any adjustment in B and the reactance B independent of
any adjustment in A, there are two ways:

a) One quantity in the numerator either Z2 or Z4 is made adjustable to bring the balance and
bridges of such type are known as ratio bridges.
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 Electrical Measurements
b) The only quantity in the denominator Z3 is made adjustable and the bridges of such type are
termed as product bridges.

Ratio Bridges:

To fulfill the above condition the ratio arms are taken such that Z2 / Z3 is real (or imaginary). If this ratio
is real then,

……………………….(3)

Where, K(real) = and as R2 and R3 are positive K must also be positive.

Thus, X2 and X3 must be reactance of same kind, i.e. both inductive or both capacitive. Also, since,
, the phase angles φ2 = φ3. From equation (1)

………………………..(4)

φ1 – φ 4 = φ 2 – φ 3 = 0

φ1 = φ 4

Hence, Z1 and Z4 must have same time constant and

R1 + jX1 = K(R4 + jX4)

When the ratio is imaginary

From which , Where K may be positive or negative.

Thus, X2 and X3 must be reactance of opposite nature, i.e. if one is inductive then the other must be
capacitive. Also

or, tan φ2 = - cot φ3 = tan (φ3 ± 900)

or, φ2 – φ3 = ± 900 …………………….. (5)

Hence, from equation (4)

φ1 – φ4 = φ2 – φ3 = ± 900

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 Electrical Measurements
Also, R1 + jX1 = K(R4 + jX4) = - KX4 + jKR4 = …………………….(6)

Which implies that X4 must be the reactance of same kind as X3, and X1 of same kind as X2. Thus, as X2
and X3 are of opposite kind X1 and X4 must also be of opposite kind.

Product Bridges:

In this case the adjustable branch is Z2 and Z2Z4 must either real or imaginary. Let

Z2Z4 = (R2 + jX2)(R4 + jX4) = K(real)

From which

R2 R4 – X 2 X 4 = K

and R2 X 4 + R 4 X 2 = 0

or,

or, tan φ2 = - tan φ4

or, φ2 + φ4 = 0

Since R2 and R4 are positive, X2 and X4 must be reactance of opposite kinds

Now, φ1 + φ3 = φ2 + φ4 = 0 ……………………………….(7)

R1 + jX1 = …………………………..(8)

This means that X1 and X3 are reactances of opposite kinds, i.e. X1 has same sign as susceptance B2. It is
important to mention here that in real product bridges the resistance in the unknown branch is
balanced by conductance in the adjustable branch and its reactance is balanced by the susceptance of
same kind. In particular, if the product arms have pure resistances, which is usual in practice, then

K = R2R4

And R1 = R2R4G3

X1 = R2R4B2 …………………………………..(9)

If the product Z2Z4 is imaginary then

Z2Z4 = (R2 + jX2)(R4 + jX4) = jK

From which,

R2 R 4 – X 2 X 4 = 0

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 Electrical Measurements
And X 2 R4 + R 2 X 4 = K

or, tan φ2 = - tan (±900 - φ4)

or, φ2 + φ4 = 900

or, φ1 + φ3 = φ2 + φ4 = 900 ……………………………….(10)

Thus, the two product branches have the same kind of impedances. From balance condition

R1 = -KB3

And X1 = KG3 …………………………………………….(11)

Measurement of Inductance:
1. Maxwell’s Bridge:
This method is very suitable for accurate measurement of medium resistances. In this method
unknown inductance is determined by comparing it with a standard self inductance.

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 Electrical Measurements
Such a bridge circuit is shown in above figure in which L1 is a unknown self inductance of resistor
R1, L3 is a known variable inductance of resistor R3 whose resistance is constant, R2 and R4 are
pure resistances and D is a detector. The magnitude of L3 should be of the same order as that of
L1. The bridge is balanced by varying L3 and one of the resistances R2 or R4. The bridge can also
be balanced by keeping R2 and R4 constant and by varying the resistance of any one of the other
two arms by connecting an additional resistance in that arm.
When the bridge is balanced, the current flowing through detector D is zero and
I 1 = I 2; I 3 = I 4
Potential difference across arm AB = Potential difference across arm AD = V1
i.e. I1Z1 = I3Z3 = V1
or, I1(R1 + jωL1) = I3(R3 + jωL3) ……………(1)
Potential difference across arm BC = Potential difference across arm CD = V2
i.e. I2R2 = I4R4 = V2
I1R2 = I3R4 ……………………………..(2)
Dividing expression (1) by (2) we have

Equating the real and imaginary parts of both sides separately we have

R1 = R3 ……………………….(3)

And

L1 = L3 ……………………….(4)
Thus the value of unknown inductance L1 can be determined.
Vector diagram for the balanced condition of the bridge is shown in figure in which currents I1
and I2 are made in phase with currents I3 and I4 by adjusting the impedances of the various
branches. Here one point is to be noted that inductances L1 and L3 should be placed at a
distances from each other and leads used in the bridge arms should be twisted together
properly to avoid loops, otherwise loops introduce inductance in the bridge circuit which may
cause error during measurement.

2. Maxwell-Wein Bridge:
In this method of measurement of self inductance the unknown self-inductance is compared
with a standard variable capacitance the circuit being shown below.

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 Electrical Measurements

In the circuit L1 is unknown self inductance and R1 is unknown resistance of the inductor, R2, R3
and R4 are known non-inductive resistances and C4 is a standard variable capacitor.
Impedance Z1 = R1 + jωL1
Impedance Z2 = R2
Impedance Z3 = R3
Impedance Z4 = = =

For balance condition of bridge

Z1Z4 = Z2Z3

R1R4 + jωL1R4 = R2R3 + jωC4R2R3R4

Equating real and imaginary quantities
R 1 R4 = R 2 R3
or, ………………………..(1)
And jωL1R4 = jωC4R2R3R4
L1 = R2R3C4 ………………………(2)
The bridge is preferably balanced by varying C4 and R4 which gives independent settings.
The Q-factor of the inductor is given by and at balance condition

Q= = ωC4R4 ……………..(3)

3. Anderson Bridge:
This method is one of the commonest and best bridge methods for precise measurement of
inductance over a wide range. In this method the unknown self inductance is measured in terms
of known capacitance and resistance by comparison. It is actually a modification of the Maxwell-
wein Bridge and is an example of a more complicated bridge network. The circuit is shown
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 Electrical Measurements
below, in which L1 is self inductance and R1 is the resistance of the coil under test, R2, R3, R4 r are
known non-inductive resistances and C is a standard known capacitor.

The bridge is preliminary balanced for steady current by adjusting R2, R3 and R4 and using an
ordinary galvanometer as detector and then bridge is balanced in alternating current by varying
r and using vibration galvanometer or telephone depending upon the supply frequency.
When the bridge is balanced
I1 = I1 ; Ir = Ic ; I3 = I4 + Ic ; V1 = V3+Icr ; V2 = V4 + Icr
From the theory of star-delta transformation
RAD = R3r

RCD = R4r

RAC = R3R4

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 Electrical Measurements

Hence the Anderson bridge is reduced to an equivalent Maxwell-wein bridge is shown in above
figure. Using the relation derived for the Maxwell-wein bridge we have

R1 = …………………………(1)

And L1 = CR2RAD = CR2R3r

L1 = ……………………..(2)

Problem

1. An ac bridge is connected as follows: Branch AB is an inductive resistor, branches BC and ED are

variable resistors, branches CD and DA are non-reactive resistors of 400Ω each and branch CE is
a condenser of 2μF capacitance. The supply is connected to A and C and the detector to B and E.
Balance are obtained when the resistance of BC is 400Ω and that of ED is 500Ω. Determine the
resistance and inductance of AB.

R2 = 400Ω, R3 = 400Ω, R4=400Ω, r = 500Ω, C = 2μF = 2 X 10-6F

R=

L=

= 2 X 10-6 X (500 (400 + 400) + 400 X 400) = 1.12 Henry

4. The Hey Bridge:

This is another modification of the Maxwell-Wein Bridge, which may be used to advantage if the
phase angle of the inductor under test is large.

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 Electrical Measurements

The circuit arrangement is shown in above figure. In this arrangement L1 is self-inductance and
R1 is the resistance of the coil under test R2, R3, R4 are known non-inductive resistances and C4 is
a standard variable capacitance. The bridge is balanced by varying R4 and C4. It is often more
convenient to use a capacitor of fixed value and to make R4 and either R2 or R3 adjustable.
When the bridge is balanced
I2 = I1; I4 = I3; V1 = V3 and V2 = V4
Since V1 = I1Z1 = I1(R1 + jωL1) and V3 = I3R3
I1(R1+jωL1) = I3R3 ……………………..(1)
And since V2 = I2R2 = I1R2
V4 = I4Z4 = I3 (R4 - )

I1R2 = I3 (R4 - ) …………………..(2)

Dividing expression (1) by (2) we have

R1 R4 + = R2R3 …………………………………(3)
Separating real and imaginaries, we have
R 1 R4 + = R2R3 ………………………………………..(4)

And, =0
Or, R1 = ω2C4L1R4 ……………………………………(5)
Solving expressions (4) and (5) we get
L1 = ………………………..(6)

And R 1 = ω2 C 4 R4 . = ………………………(7)

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 Electrical Measurements
Expression (6) and (7) indicates that the balance condition is a function of frequency. However,
the frequency need not be accurately known to determine inductance since ω appears only in a
term which will be small compared to unity in cases where use of Hey bridge is indicated (i.e.
where Q factor of coil is large).

Q-factor of the coil = =

Now the term in the denominators of equation (6) and (7) has the value of 0.01 if
Q=10 and even more smaller for higher values of Q so the term can be dropped
without causing an appreciable error. In case, this term is to be included in calculations of L1 and
R1 then it is of such minor importance that it may be computed with sufficient accuracy from an
approximate value of frequency. Here it should be noted if the term is excluded from
the equation, then it is same for L1 as for the Maxwell Bridge.
L1 =

But Q=

So, L1 =

For the value of Q greater than 10, the term will be smaller than and so can be
neglected.
Therefore L1=R2R3C4 and it is the same as for the Maxwell Bridge.
The bridge has the advantage of requiring only a relatively low-value resistor for R4 where as for
large inductance low resistance coils, the Maxwell-wein would require a parallel resistance R4 of
very high value, perhaps 105 or 106Ω. Resistance boxes of such high values are very costly.
This bridge is not suited for the measurement of low Q-factor of the inductors, because in that
case, the term in the denominator becomes very important. And then it is required to
know the bridge frequency to a value accurate limit. Moreover with low value of Q-factor, it
gives poor convergence in balancing.

Problem:

1. The four arms of a Hey bridge are arranged as follows: AB is a coil of unknown impedance, BC is
a non-inductive resistor of 1000Ω, CD is a non-inductive resistor of 833Ω in series with a
standard capacitor of 0.38μF. DA is non-inductive resistor of 16800Ω. If the supply frequency is
50Hz, determine the inductance of the resistance at the balanced condition.

R2 = 1000Ω; R3 = 16800Ω; R4 = 833Ω; C4 = 0.38μF;

L1 = = = 6.321 H.

R1 = = = 197.487Ω

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Gautam Sarkar, Electrical Engineering Department, Jadavpur University
 Electrical Measurements
5. The Owen Bridge:
The bridge circuit also determines the unknown inductance in terms of resistance and
capacitance. The advantage of this method is that the inductances over a wide range can be
determined by this method by employing.

The arrangement of the bridge as shown in above figure in which the unknown inductance L1
whose resistance is R1 is connected in series with another variable non-inductive resistor R. The
capacitance C2 is also connected in series with a non-inductive resistor and C4 is known standard
capacitor. The bridge is balanced by successively varying R1 and R2 in the circuit.
At balance

Or, (R1 + jωL1 + R) = R3 (R2 - )

Equating real and imaginary quantities, separately
L1 = R2R3C4
And R1 =
Since the expression obtained for unknown quantities R1 and L1 does not have ω, so the
balancing of bridge is independent of frequency and wave form.

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Gautam Sarkar, Electrical Engineering Department, Jadavpur University
 Electrical Measurements
Measurement of Capacitance:
1. De Sauty’s Bridge:
This bridge is used to determine the unknown capacitance by comparing it with known standard
capacitor. The circuit is shown in below figure in which C1 is a standard capacitor of known
magnitude and R1 and R2 are known non-inductive resistances. The bridge is balanced by
adjustment of either R1 or R2.

At balance I 1 R1 = I 2 R2
And

Then,

Or, C1 =
This is very simple method but it is very difficult rather impossible to obtain a perfect balance if
both of the capacitors are not free from dielectric loss. Only in the case of air capacitors a
perfect balance can be obtained.
For computing two imperfect capacitors the bridge is modified by connecting resistances in
series with them. R3= and R4 are the series resistances, where r1 and r2 are small resistances
representing the loss components of the capacitors. Balance is obtained by variation of the
resistances R1, R2, R3, R4.

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 Electrical Measurements

At balance I 1 R1 = I 2 R2
And I1 (R3+r1 ) = I2(R4+r2 )

Then

Or, R1(R4+r2) - = R2(R3+r1) -

Equating real and imaginary quantities separately
R1(R4+r2) = R2(R3+r1)
And

Or,

The angles δ1 and δ2 are the phase angles of capacitors C1 and C2 respectively.
δ1 = tan δ1 = = r1ωC1

δ2 = tan δ2 = = r2ωC2 (δ1 and δ2 are being very small)

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 Electrical Measurements
δ2 – δ1 = ω(r2C2 - r1C1) = ω(C1R3 - C2R4) = ωC1(R3 - )
From the above expression the phase angle of one capacitor can be determined in terms of the
phase angle of the other one.

Problem:

1. In a modified De Sauty bridge measurement the following readings are obtained: R1=1000Ω;
R2=1000Ω; R3=2000Ω; R4=2000Ω; C1=1μF; f=1000Hz; r=10Ω (res. Of C1). Find out phase angle
errors and unknown capacitance.

R1=1000Ω; R2=1000Ω; R3=2000Ω; R4=2000Ω; C1=1X10-6 F;

ω = 2πf = 2 X 3.14 X 1000 = 6.280; r1=10Ω;
Unknown Capacitance C2 = = 1 x 10-6 X = 1 X 10-6 F = 1μF

r2 = = = 10Ω

Phase angle error, δ1= r1ωC1 = 6280 X 10 X 1 X 10-6 rad = 6280 X 10 X 1 X 10-6 X = 3.60
Similarly, δ2 = r2ωC2 = 6280 X 10 X 1 X 10-6 rad = 6280 X 10 X 1 X 10-6 X = 3.60

2. Schering Bridge:
The circuit is shown in figure in which C1 is the capacitor under test, R1 is an imaginary resistance
representing its dielectric loss component; C3 is a standard capacitor. C4 is a variable capacitor
and R2 and R4 are known non-inductive resistors. The resistance R2 is variable.

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 Electrical Measurements
The impedance of the Schering Bridge arms are:
Z1 = = =

Z 2 = R2
Z3 =

Z4 =

Under balanced conditions

Z1Z4 = Z2Z3

i.e. . =

or, . =

or, =

Equating real terms we have

or, =

From vector diagram for the capacitor C1 and R1 are in parallel

Cos δ = = = =

or, Cos2 δ =

Substituting value of = Cos2 δ

C1 =

Again at balance condition

tan δ = = ωC4R4

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 Electrical Measurements
and also tan δ =

ωC4R4 =

R4 =

C1 =

Where δ is known as a loss angle of capacitor and sin δ is known as a power factor of the
capacitor.

3. High Voltage Schering Bridge:

This is one of the most important and useful bridge circuits available for measurement of
capacitance, dielectric loss of a capacitor and power factor of cables. It is generally used, both
for precision measurements of capacitors at low voltage and for study of insulation and
insulating structures at high voltages.
The bridge is also used for measurement of capacitance and dissipation factor for insulation and
in determining the general properties of condenser, bushings, insulating oil and other insulating
materials which in any case have very small capacitances, so for such types of measurements
high voltage Schering bridge as shown in figure required. But certain modifications have to be
made in Schering Bridge to operate at high voltage.

1. In this case a high voltage supply is obtained from a transformer usually at the power
frequency and the detector used is vibration galvanometer.
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Gautam Sarkar, Electrical Engineering Department, Jadavpur University
 Electrical Measurements
2. The Schering Bridge is advantageous for high voltage tests due to the fact that the arms AB
and AD, each contains only capacitors, which can be specially constructed for high voltage
work. Also these two arms will have very high impedances, at power frequencies in
comparison with that of the arms BC and CD. Thus the major portion of the source potential
difference exists across the arms AB and AD and the potential difference across the arms BC
and CD, which contains the control, is small. In order to afford safely to the operator from
the high voltage hazards, while carrying out control, junction point C is earthed.
The arrangement is ideal from the point of view of safety but it is much less sensitive than
its conjugate (with the source and detector interchanged). However at high voltages
sensitivity, with a good detector will usually be ample for all practical purposes. In precise
measurements at low voltages the conjugate bridge is often used since in this case
sensitivity will be much greater.
3. In order to prevent dangerous rises of voltage across these arms in case of breakdown of
either of the high voltage capacitors; a spark (set to breakdown at about 100 volts) is
connected across each of the arms BC and CD.
4. The impedances of arms AB and AD are high so current from the source is less. The
arrangement is ideal from the point of view of power consumption but it is much less
sensitive so a sensitive detector has to be used.
5. In operating the Schering Bridge at high voltages and in the precise measurements at low
voltages it is extremely important that the effect of stray capacitances between the bridge
elements be eliminated. This is done by enclosing the vulnerable portions of the network
within electrostatic shields and by maintaining these shields at suitable potentials. Such a
shielded bridge suitable for high voltage operation is shown in figure. Earth capacitance
effect on the galvanometer and leads is eliminated by using Wagner earth device.
6. A capacitor C3 of fixed value is used which has normally air or compressed gas as dielectric.
The dissipation factor with dry and clean gas is practically zero but loss in the insulating
supports cannot be avoided. However such losses can be prevented from influencing the
measurement by the use of a guard ring from which both the high and low voltage
electrodes are supported. Then the current through the high voltage supports passes direct
to earth and as the potential difference between the low voltage electrode and the guard
ring is very small, the insulation of the low voltage electrode has only a very small effect. In
case, solid dielectric standard capacitor is used then the losses must be accurately known.

Measurement of Frequency:
1. Wien’s Bridge:
The Wien’s Bridge is primarily known as a frequency determining bridge. A Wien’s bridge
may be employed in a harmonic distortion analyzer, where it is used as notch filter,
discriminating against one specific frequency. The Wien’s bridge also finds applications in
audio and HF oscillators as the frequency determining device,

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Gautam Sarkar, Electrical Engineering Department, Jadavpur University
 Electrical Measurements

In the above figure shows a Wien’s bridge under balance conditions.

At balance =

or,
Equating the real and imaginary parts,
……………………………………(1)

and

From which ω=

And frequency f= Hz …………………………..(2)

In most Wien bridges, the components are chosen that R1 = R2 = R and C1 = C2 = C
Then equation (1) reduces to

And equation (2) reduces to f=

Switches for resistors R1 and R2 are mechanically linked so as to fulfill the condition R1=R2. As
long as C1 and C2 are fixed capacitors equal in value and R4=2R3, the Wien’s bridge may be
used as a frequency determining device, balanced by single control. This control may be
directly calibrated in terms of frequency as is evident from equation (3).
This bridge is suitable for measurement of frequencies from 100Hz to 100 KHz. It is possible
to obtain an accuracy of 0.1% to 0.5%.

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Gautam Sarkar, Electrical Engineering Department, Jadavpur University
 Electrical Measurements
Apparatus Used in Conjunction with AC Bridges:

Now it has become clear that the apparatus required for the construction and use of AC bridges
may be classified under the following three headings:

1. Standards of resistance, inductance and capacitance.

2. Sources of alternating current
3. Detectors.

Sources of alternating current: These may be divided into the following classes:

(a) Interrupters
(b) Microphone Hummers
(c) Alternators
(d) Oscillators

(a) Interrupters:

An interrupter is a very common source of alternating current used for bridge measurements.
This consists of a small induction coil or transformer and an interrupter. The primary circuit of
the transformer contains a battery and an interrupter working at a constant frequency supplies
the current to the bridge.

A simple example of interrupters may be an electrically maintained tuning fork of suitable

frequency, operating a contact which regularly opens or closes the primary circuit.

The frequency depends on the pitch of the fork and is therefore very constant. The waveform at
the secondary terminals is fairly pure. Frequencies which may be obtained lie between 50Hz and
about 1000Hz. The disadvantage of interrupters is the small amount of power available.

(b) Microphone Hummers:

The principle utilized in a microphone hummer is that when a microphone, connected in series
with a battery is subjected to suitable mechanical vibrations the current through it will be made
to vary. This varying current can be fed back in the form of vibrations, produced by the feedback
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 Electrical Measurements
system, to maintain the oscillation of the microphone granules. Thus this principle is similar to
that applied in an oscillator.

For an example of the microphone hummers,

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