1. A.C. Bridges

2. The inductance and capacitance are measured accurately with the help ofA.C.Bridges.
For operation of the A.C. bridges, A.C. source is required.
For detection of the balanced condition of bridge VIBRATION GALVANOMETER, CRO, HEADPHONES are used.
Headphones are used between range 250 Hz to 4 KHz.
In A.C. Bridges there are several types in it :
Basic A.C. Bridge
Maxwell's Bridge
Hay's Bridge
Anderson Bridge
Schering Bridge
Heaviside Bridge
Wien Bridge
Owen’s Bridge
De-Sauty’s Bridge
Carey-Foster Bridge

3. 2.1.1 Basic A.C.Bridge :
As the name indicates ac bridges consists of an ac supply. This provides ac voltage at the required frequency. The null detector is used to give indication of unbalanced. This gives response to the unbalanced current in the bridge circuit. The simplest form of AC Bridge as shown in following fig.:

4. In the simplest form an ac bridge consists of four impedances
In the simplest form an ac bridge consists of four impedances. The headphone is used as a null detector. In some cases ac amplifier with an output meter is used as null detector.
The null response is obtained by adjusting one of the bridge arms.
The null indication is obtained when the voltage at point B is equal to the voltage at point D. These two voltages should be equal in terms of both amplitude and phase. That means to obtain this condition; the voltage drop across AB should be equal to the voltage drop across AD.
Eab=Ead
In terms of current and impedance we can write,
I1Z1=I2Z
Now for the null indication, there should not be the flow of current from point B to point D. Thus under balancing condition of bridges, current I1 passes through Z1 and Z3. While current I2 passes through Z2andZ4. Thus we can write equation for currents as,
and

5. Putting Equations (3) and (4) in equation (2) we get,
(5)
Equation (5) gives the balancing condition in terms of impedance. But in the terms of admittance Equation (5) can be written as,
Y1Y4=Y2Y3
Equation (5) indicates that the product of impedances of one pair of opposite arms is equal to the product of impedance of the pair of opposite arms. If we will consider both magnitude and phase of impedance then we have,
Here,
Z= Complex notation
Z'= Magnitude of complex impedance
= Phase of complex impedance
Thus it can be written as,
(6)
From equation (9) we can conclude that there are two condition of balancing the Ac Bridge. These are as follows:
The product of the magnitudes of opposite arms must be equal. That means
Z’1Z’4=Z’2Z’3
The sum of phase angles of opposite arms must be equal. That means

6. Maxwell’s Bridge:
The Maxwell’s bridge measures an unknown inductance in term of a known capacitance.
Fig shows the schematic diagram and fig shows phasor diagram.
One of the ratio arms has a resistance and capacitance in parallel. We known that the general equation of bridge balance is,
Z1Zx=Z2Z3
Zx=Z2Z3 X 1/Z1
= Z2Z3Y1
(Y1=Admittance of arm 1)
Z2=R2, Z3=R3
And
Y1=1/R1 +JwC1
Substituting these value,
Zx=Rx+JwLx=R2R3 (1/R1+JwC1)
Zx=R2R3/R1+JwR2R3C1
Separating real and imaginary parts,
Rx=R2R3/R1
Lx = R2R3C1
To obtain bridge balance, first R3 is adjusted for inductive balance and R1 is adjusted for resistive balance.
The quality factor of the coil is given by,
Q=WLx/Rx
=WR1R2C1/R2R3/R1
Q=W R1 C1

8. ADVANTAGES:
1) The balance equation is independent of frequency.
2) The two balance equations are independent.
3) The scale of resistance can be calibrated to read the inductance and Q value.
4) It is useful for measurement of wide range of inductance at power and audio frequencies.
DISADVANTAGES:
1)It cannot be used for measurement of high Q values, it is limited to measure low Q value (1<Q<10)
2) It cannot be used for measurement of very low Q values,because of balance converge problem.
Commercial Maxwell’s bridge measures the inductance from H, with2% error.

9. 2.1.3 Hay’s Bridge :
The limitation of Maxwell’s bridge is that it can be used for high Q values.The Hay’s bridge is suitable for the coils having high Q values.
The different in Maxwell’s bridge and Hay’s bridge is that the Hay’s bridge consists of resistanceR1 in series with the standard capacitor C1 in one of the ratio arms. Hence for larger phase angles R1needed is very low, which is practicable. Hence bridge can be used for coils with high Q values.

11. ADVANTAGES:
1) It is best suitable for the measurement of inductance with high Q, typically greater than 10.
2) It gives very simple expression for Q factor in terms of elements in the bridge.
It requires very low value resister R1 to measure high Q inductance.
DISADVANTAGES:
1) It is only suitable for measurement of high Q inductance. Consider expression for unknown inductance.

12. 2.1.4 Anderson’s Bridge
The Anderson’s bridge is a modification over the Maxwell’s bridge. In Anderson’sbridge, the self inductance is measured in terms of a standard capacitor.fig shows the schematic for Anderson’s bridge.

14. We known the condition for bridge balance is,
ZxZ4=Z2Z3
At balance,
I1 = I3and I2=IC + I4.
I1R3=IC X 1/JwC
IC = I1 JwCR3
The other balance equations are,
I1 (Rx+r1+JwLx)=I2R2 + ICR.
IC(R+1/Jwc)=(I2 – IC) R4
Substituting the value of IC in equation we get,
I1(Rx+r1+JwLx)= I2R2+I1 JwCR3R
I1(Rx+r1+JwLx)= I2R2
Substituting the value of Ic in equation we get,
JwCR3I1(R+1/Jwc)=(I2 – I1 JwCR3)R4
I1(JwR3R + R3) + I1 JwCR3R4 = I2R4
I1(JwCR3R + R3+JwR3R4) = I2R4
I1(JwCR3R + R3 + JwCR3R4 /R4)=I2
Substitution the value of I2 in equation, we get
I1(Rx + r1 + JwLx – JwCR3R)=(JwCR3R +R3+JwCR3R4/R4)R2 I1
Equation real and imaginary parts,
Rx = R2R3/R4 – r1
Lx = C R3/R4{r(R4 + R2) + R2R4}
ADVANTAGES:
1) In case of Anderson‘s bridge balance is easily obtained for low Q coils
2) The bridge can be used for accurate determination of capacitance in terms of inductance.
DISADVANTAGES:
1)The Anderson’s bridge is complex.
2) The bridge balance equationis not simple. They are rather more tedious.
3) The addition of junction increases the difficulty of shielding of the bridge.

15. Wien Bridge It is a bridge used for the measurement of frequency.
Apart from this it has a variety of applications in the harmonic distortion analyzer where it is used as a notch filter,discriminating against a specific frequency in the audio and HF oscillators as frequency determining against a specific frequency in the audio and HF oscillators as frequency determining element.
Fig shows the circuit.

17. It consists of an ac series circuit in one armR1C1 AND a RC parallel combination in the adjoining arm.
The bridge balance equation is
Z1Z4=Z2Z3
Z1 =R1 -
Z2=R2
Z3=R3C3 parallel combination
Y3 =
Z4 =R4
Substituting,
Z2=Z1Z4Z3
R2 =(R1- )
R2 =(
Equal real terms
R2=
Equating imaginary terms, j
Equation gives resistance ratio, while equation gives frequency
R1=R3=R
And
C1=C3=C
Equation gives resistance ratio, gives frequency,
Normally
R1=R2=R
C1=C2-=C
Above equation is general equation for determine the frequency of a Wien bridge.
Practically C1 and C3 are fixed capacitors and R1 and R3 are variable resistors which are controlled by a common shaft.
As long as, the bridge can be used as frequency determine device.
It is suitable for measurement of frequency from 100 HZ to 100KHZ, within accuracy of 0.1 to0.5 percent.
It also used for the measurement of capacitance.

18. De-Sauty’s Bridge :

19. It is the simplest method for capacitance measurement fig
It is the simplest method for capacitance measurement fig. shows the basic circuit arrangement.
We know that equation for bridge balance is,
Z1 Zx =Z2 Z3
Z1 = R1,
Zx =
Z2 = R2,
Z3=
R1 = R2
C3 = Cx
By changing the values of R3 or R4, bridge can be easily obtained

20. ADVANTAGE:
1. The bridge is simplest
2. It is economical
DISADVANTAGE:
1. If both the capacitor are not free from dielectric loss, then it is not possible to achieve bridge balance. This method is only suitable for the measurement of lossless capacitor e.g. air capacitors.

21. Carey-Foster Bridge :

22. This bridge is used for the measurement of capacitance in terms of standard mutual inductance.
The same bridge can be used for measurement of mutual inductance in term of standard capacitance. Then it is known as Heyd Weilling Bridge.
The circuit diagram and phasor diagram is as shown in fig. explains the connection of various components used for the bridge.
One of the bridge arms (e-d) is a short circuited arm therefore the potential drop across the arm is zero.
Thus when the balanced is achieve, the potential drop across the arm (a-b) should also be equal to zero.
Therefore a negative coupling is needed for the mutual inductance
When the balance is reached the equation can be written as :
I1(R1+j
And
I1(R3+ ) = I1

23. By solving these equation we get
M = R1×R4×C3
It the bridge is used for the measurement of capacitance the unknown capacitance C3 can be expressed as:
C3 =
And the series resistance component of the capacitance R1 is given by
R3 =
If the bridge is used to measurement of mutual inductance then the resistance R3 will be a separate standard capacitance like C3 can be used.
Equivalently if a standard loss less capacitance is not available then R3 represent the equivalent series resistance of capacitor C3.