A.C.

Bridges-

Alternating current bridge methods are of outstanding importance for measurement

of electrical quantities. Measurement of inductance, capacitance, storage factor, loss factor
may be made conveniently and accurately by employing ac bridge networks.

The AC bridge is a natural outgrowth of the Wheatstone bridge. An ac bridge, in its

basic form, consists of four arms, a source of excitation, and a balance detector. In an ac
bridge, each of the four arms is an impedance, and the battery and the galvanometer of the
Wheatstone bridge are replaced respectively by an ac source and a detector sensitive to
small alternating potential differences.

The usefulness of ac bridge circuits is not restricted to the measurement of unknown

impedances and associated parameters like inductance, capacitance, storage factor,
dissipation factor etc. These circuits find other applications in communication systems and
complex electronic circuits. Alternating current bridge circuits are commonly used for
phase shifting, providing feedback paths for oscillators and amplifiers, filtering out
undesirable signals and measuring frequency of audio signals.

For measurements at low frequencies, the power line may act as the source of supply
to the bridge circuits. For higher frequencies, electronic oscillators are universally used as
bridge source supplies. These oscillators have the advantage that the frequency is constant,
easily adjustable, and determinable with accuracy. The waveform is very close to a sine
wave, and their power output is sufficient for most bridge measurements. A typical
oscillator has a frequency range of 40 Hz to 125 kHz with a power output of 7 W.

The detectors commonly used for ac bridges are:

(i) Headphones,

(ii) Vibration galvanometers, and

(iii) Tuneable amplifier detectors.

Headphones are widely used as detectors at frequencies of 250 Hz and over upto 3
or 4 kHz. They are most sensitive detectors for this frequency range. When working at a
single frequency a tuned detector normally gives the greatest sensitivity and discrimination
against harmonics in the supply.

Vibration galvanometers are extremely useful for power and low audio frequency
ranges. Vibration galvanometers are manufactured to work at various frequencies ranging
from 5 Hz to 1000 Hz but are most commonly used below 200 Hz as below this frequency
they are more sensitive than the headphones.
 Tuneable amplifier detectors are the most versatile of the detectors. The transistor
amplifier can be tuned electrically and thus can be made to respond to a narrow bandwidth
at the bridge frequency. The output of the amplifier is fed to a pointer type of instrument.
This detector can be used, over a frequency range of 10 Hz to 100 kHz.

General Equation for AC Bridge Balance: -

The conditions for the balance of bridge require that there should be no current
through the detector. This requires that the potential difference between points b and d
should be zero. This will be the case when the voltage drop from a to b equals to voltage
drop from a to d, both in magnitude and phase. In complex notation we can, thus, write:

So, we can write the above-stated condition as,

E1 = E2
Applying ohms’ law
I1 Z1 = I2 Z2
At balance,

And
Substituting the value of I1 and I2-

Z1 (Z2 + Z4) = Z2 (Z1 + Z3)

Z1Z2 + Z1Z4 = Z1Z2 + Z2Z3

Hence,

Z1Z4 = Z2Z3

Or when using admittances instead of impedances-

Y1 Y 4 = Y 2 Y 3

Considering the polar form, the impedance can be written as Z = Z ∠θ, where Z
represents the magnitude and θ represent the phase angle of the complex impedance.
Now that equation can be re-written in the form

(Z1∠θ1)(Z4∠θ4) = (Z2∠θ2)(Z3∠θ3)

Thus, for balance, we must have,

Z1 Z4 ∠θ1 + ∠θ4 = Z2 Z3 ∠θ2 +∠ θ3

The above equation shows that two conditions must be satisfied simultaneously when
balancing an ac bridge. The first condition is that the magnitude of impedances satisfies
the relationship :

Z1 Z4 = Z2 Z3

The second condition is that the phase angles of impedances satisfy the relationship:

∠θ1 + ∠θ4 = ∠θ2 +∠ θ3

 The phase angles are positive for an inductive impedance and negative for
capacitive impedance.

If we work in terms of rectangular coordinates, we have-

Z1= R1 + jX1 ; Z2= R2 + jX2

Z3= R3 + jX3 and Z4= R4 + jX4

For balance, Z1 Z4 = Z2 Z3

or (R1 + jX1) (R4 + jX4) = (R2 + jX2) (R3 + jX3)

R1 R4 – X1 X4 + j(X1R4 + X4R1) = R2 R3 – X1 X4 + j(X2R3 + X3R2)

The above equation is a complex equation and a complex equation is satisfied only
if real and imaginary parts of each side of the equation are separately equal. Thus, for
balance,

R1 R4 – X1 X4 = R2 R3 – X1 X4

X1R4 + X4R1 = X2R3 + X3R2

Thus, there are two independent conditions for balance and both of them must be
satisfied for the ac bridge to be balanced.
MEASUREMENT OF INDUCTANCE-
1. MAXWELL'S INDUCTANCE BRIDGE: -

This bridge is used to measure self-inductance by comparison with a variable standard

self-inductance. The connection of bridge and phasor diagram for balanced circuit is
shown:

r2

Here,

L1 = inductance to be measured with resistance R1

L2=variable inductance
R2=variable resistance
R3, R4= pure resistance
Under balanced condition (i.e., when detector shows null deflection), we have,

On equating the real and imaginary parts on both sides, we get,

Hence, the unknown self-inductance and resistance of the inductor are obtained in terms
of known standard values. Also, both the equations are independent of frequency term.
2. Maxwell’s Inductance Capacitance Bridge:

Here, an inductance is measured by comparison with standard capacitance. The connection

of bridge and phasor diagram for balanced circuit is shown.
Here,

L1= inductance to be measured with resistance R1

R2, R3, R4= are known pure resistance.

C4=variable standard capacitor.

Now,
Z1=(R1+jωL1);
Z2=R2;
Z3=R3;
Z4=R4/1+jωC4R4

The balanced condition is that,

Z1Z4=Z3Z2

(R1+jωL1)(R4/1+j𝜔C4R4)=R2R3

R1R4+jωL1R4=R2R3+jωR2R3C4R4

Equating real and imaginary part of both sides,

𝑅2𝑅3
𝑅1 =
𝑅4

𝐿1 = 𝑅2𝑅3𝐶4

Expression for Q factor,

Q =ωL1/R1

=ωC4R4

Advantages-

• Equation of L1 and R1 are simple.

• They are independent of frequency.
• They are independent of each other.
• Standard capacitor is much smaller in size than standard inductor.

Disadvantages-

• Standard variable capacitance is costly.

• It can be used for measurements of Q-factor in the ranges of 1 to 10.
• It cannot be used for measurements of choke with Q-factors more than 10.
For measuring chokes with higher value of Q-factor, the value of C4 and R4 should be
higher. Higher values of standard resistance are very expensive. Therefore, this bridge
cannot be used for higher value of Q-factor measurements.

3. HAY'S BRIDGE:

Hay's bridge is a modification of Maxwell's bridge with a series resistance connected to

standard capacitor, instead of resistance parallel with capacitor.

Let,

L1= inductance to be measured with resistance R1

R2, R3, R4= are known pure resistance.

C4= Standard capacitor.

Now,

Z1=R1+jωL1;

Z2=R2;

Z3=R3;

Z4=R4−j/ωC4
Equating real and imaginary terms on both sides, we get,

Substituting equation 2 in 1, we get,

Substituting equation 3 in 2, we get,

Now, the quality factor of an inductor is given by,

Substituting equation 5 in 3, we have,

For high Q coils i.e., Q > 10, 1/Q2 is almost negligible. Hence the above equation reduces
to,

From the above equations, we can say that for high Q coils the expression for L1 is
free from the frequency term. For low Q coils, 1/Q2 cannot be neglected and hence to find
L1, the frequency of source is to be accurately known. Therefore, the bridge suits only for
the measurements of inductance of high Q coils.

Advantages of Hay's Bridge:

• The expression obtained for the Q-factor of the coil using Hay's bridge is not a
complicated one.

• From the above expression, it can be seen that the resistance R4 is inversely
proportional to the Q-factor. Lower the resistance higher the Q-factor. Thus for high
 Q coils, the value of resistance R4 should be quite small. Hence, the bridge requires
the resistance of low value.
• Hay's bridge is suitable for coils whose quality factor is greater than 10 (Q > 10).
Also, it gives a simple expression for unknown inductance for high Q coils.

Disadvantages of Hay's Bridge:

• The major drawback of Hay's bridge is that it cannot be used for measuring coils
having a Q-factor less than 10.

MEASUREMENT OF INDUCTANCE-
1. Desauty's Bridge:

A null indicator is connected across terminals B and D, which indicates null deflection
when the bridge is balanced.

Let,

C1 = Unknown capacitance

C2 = Known standard capacitance

R3 = Known standard non-inductive resistance

R4 = Known standard non-inductive resistance.

From the above figure, the impedances in each arm is given as,

When the bridge is balanced, we have,

It is the easiest method for the measurement of capacitance as it has a simple

circuit and only one variable element. In order to bring the bridge into a balanced
condition, either R3 or R4 can be chosen as a variable element.

Modified De sauty's Bridge:

The modified De sauty's bridge is used for the measurement of unknown imperfect
capacitance. It is the modification of De sauty's bridge. Imperfect capacitance is those
which contain dielectric losses. The circuit diagram of the bridge is shown below.
 In modified De sauty's bridge, the two capacitors C1 and C2 are connected in series
with two resistors R1 and R2. To measure capacitance with dielectric losses another two
resistors r1 and r2 are connected to the two capacitors that give the loss component of their
respective capacitor.

Let,

C1 = Unknown imperfect capacitance

C2 = Known standard capacitance

R1, R2, R3, R4 = Known variable non-inductive resistances

r1, r2 = Resistances having loss component of two capacitors.

From the above figure, the impedances in each arm is given as,
 When the bridge is balanced, we have,

Equating real terms on both sides, we get,

Equating imaginary terms on both sides, we get,

Comparing equations 1 and 2, we get,

The balanced condition in modified De sauty's bridge is determined by changing the

resistances R1, R2, R3, and R4 in the circuit. The dissipating factor D of the two capacitors
are,

Simplifying equation 3, we get,

Also from equation 3, we have,

In the above equation, if one of the dissipation factors is given, the other dissipation
factor can be determined. One of the disadvantage of this method is that the dissipation
factor is not precise and accurate because the differences in their resistance quantities (i.e.,
R1 R4/R3 and R2) are very small.

Hence, the difference is not highly accurate and so the dissipating factor (D) is not
used for accurate measurements.

2. Schering Bridge-

Schering bridge is an ac bridge used for the measurement of capacitance, dielectric

loss, and power factor of an unknown capacitor. It is the most popular and widely used
method for measuring capacitance. The principle of operation is similar to other ac bridges
i.e., balancing the bridge and comparing the unknown value with a known value.
 The circuit connection of a low voltage Schering bridge is as shown below. The arm
AB consists of capacitor C1 in series with resistor R1. The resistor R1 represents the
dielectric loss component of the capacitor C1.

The capacitor C2 is the standard capacitor and C4 is a variable one. The resistance
R3 is a non-inductive resistance and R4 is a variable non-inductive resistance connected in
parallel with a capacitor C4. The supply is connected between nodes A and C while the
detector is connected between B and D.

From the above figure, the impedances of the arms of the bridge are,

Under balanced condition of the bridge, we have,

Equating real and imaginary terms, we get,

Now the dissipation factor of the capacitor under measurement is,

Generally, C2 and R4 are kept constant and balance is obtained by varying R3 and
C4, so that the dissipation factor can be directly measured in terms of C4.

Limitations of Low Voltage Schering Bridge:

• The calibration for the dissipation factor is useful only for one value of frequency.
In order to use it for other frequencies, a correction has to be made by multiplying
with the frequency ratio.
• The detector used is not so sensitive.
• It is quite difficult to obtain a balanced condition.
• Errors are present in the measurements of small capacitances using a low voltage
Schering bridge.
MEASUREMENT OF FREQUENCY

Wien's Bridge-

The Wien’s bridge use in AC circuits for determining the value of unknown
frequency. The bridge measures the frequencies from 100Hz to 100kHz. The accuracy of
the bridges lies between 0.1 to 0.5 percent. The bridge is used for various other applications
like capacitance measurement, harmonic distortion analyzer and in the HF frequency
oscillator.

When the bridge is in the balanced condition, the potential of the node B and C are
equal, i.e., the V1 = V2 and V3 = V4 The phase and the magnitude of V3 = I1R3 and V4
= I2R4 are equal, and they are overlapping each other. The current I1 flowing through the
arm BD and the current I2 flowing through R4 is also in phase along with the I1R3 and
I2R4.

The total voltage drop across the arm AC is equal to the sum of the voltage drop
I2R2 across the resistance R2 and the capacitive drop I2/wC2 across the capacitance C2.
When the bridge is in a balanced condition, the voltage V1 and V2 both are equals in
magnitude and phase.

The phase of the voltage V1 and the voltage drop IRR1 across the arms R1 is also
same. The resistance R1 is in the same phase as that of the voltage V1. The phasor sum of
V1 and V3 or V2 and V4 will give the resultant supply voltage.
At balance condition,

On equating the real part,

On comparing the imaginary part,

By substituting the value of ω = 2πf,

The slider of the resistance R1 and R2 mechanically connect to each other. So that, the
R1 = R2 obtains.