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# Wien Bridge and Resonance Bridge

### Wien Bridge:

Wien bridge is primarily popular as a frequency determining bridge. It has several applications. Some of the applications are in harmonic distortion analyser, in audio frequency generation. It has a series RC combination in the adjoining arm. The other two arms are provided with resistances. The impedance of arm 1 is Z1 = R1 – j/ωC1. The admittance of arm 3, is Y3 = 1/R3 + j ωC3. The arrangement is shown in figure.

The balance equation can be obtained as follows:

R2 = (R1 – j/ωC1) R4(1/R3+jωC3) ------------------------- 1

Expanding the expression we have:

R2 = R1R4/R3 + (jωC3R1R4) – jR4/ ωC1R3 + R4C3/C1------------------------- 2

Equating the real terms:

R2 = R1R4/R3 + R4C3/C1 ------------------------- 3

It can be reduced to:

R2/R4 = R1/R3 +C3/C1 ------------------------- 4

Equating the imaginary terms:

ωC3R1R4 = R4/ ωC1R3 ------------------------- 5

where, ω = 2πf

Solving for f, we get

f = 1/2π ( C1C3 R1R3) ------------------------- 6

From the above we observe that the two conditions for balance result in an expression determining the required resistance ratio, R2/R4 and another expression determining the frequency of the applied voltage. So we have to satisfy the above equation (4) and (6), to balance the bridge.

The arrangement in most of the wien bridge circuits is such that the values of R1, R2 and C1,C3 are made equal. Hence the equation (4) reduces to R2/R4 = 2. It also reduces equation (6) to

f = 1/2 πCR ------------------------ 7

Thus Equation (7) is the general expression for the frequency of the Wien Bridge.
Practically Capacitors C1 and C3 are fixed values. R1 and R3 are variable resistors controlled by a common shaft. Now providing R2 = 2R4, the bridge can be used as a frequency determining device with single balance control, which can be calibrated directly in terms of frequency. The source supplying this bridge must be free from harmonics. If not, the balancing will be difficult. Hence it is clear that the bridge is frequency sensitive.

### Resonance Bridge:

Resonance bridge consists of reactance concentrated in one arm. They are adjusted to give series resonance so that this arm offers resistance impedance. The resonance bridge is shown in figure. From the schematic diagram of the bridge, we find that the ratio arms are formed by R1 and R2. Resistance R3 is connected in the standard arm. The fourth arm consists of an inductance Lx, capacitance Cx and resistance Rx.

This bridge can be used to measure frequency in terms of inductance and capacitance. It is also used to measure capacitance in terms of frequency and a variable inductance. It can also be used to measure inductance in terms of frequency and a variable capacitance. The balance equation can be obtained as follows:

R1(Rx + jωLx – j/ωCx) = R2 R3

At resonance

XL = Xc and fx = 1/2π (LC)

Zx = Rx

Therefore, Rx = R2 R3/R1

As can be seen from the above the bridge is balanced by resistance alone. Resistance R3 is used for this purpose.