###
**Wien Bridge:**

_{1}= R

_{1}– j/ωC

_{1}. The admittance of arm 3, is Y

_{3}= 1/R

_{3}+ j ωC

_{3}. The arrangement is shown in figure.

The balance equation can be obtained as follows:

R

_{2}= (R_{1}– j/ωC_{1}) R_{4}(1/R_{3}+jωC_{3}) ------------------------- 1
Expanding the expression we have:

R

_{2}= R_{1}R_{4}/R_{3}+ (jωC_{3}R_{1}R_{4}) – jR_{4}/ ωC_{1}R_{3}+ R_{4}C_{3}/C_{1}------------------------- 2
Equating the real terms:

R

_{2}= R_{1}R_{4}/R_{3}+ R_{4}C_{3}/C_{1}------------------------- 3
It can be reduced to:

R

_{2}/R_{4}= R_{1}/R_{3}+C_{3}/C_{1}------------------------- 4
Equating the imaginary terms:

ωC

_{3}R_{1}R_{4 }= R_{4}/ ωC_{1}R_{3}------------------------- 5
where, ω = 2πf

Solving for f, we get

f = 1/2π √( C

_{1}C_{3}R_{1}R_{3}) ------------------------- 6
From the above we observe that the two conditions for balance result in an expression determining the required resistance ratio, R

_{2}/R_{4}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 R

_{1}, R_{2}and C_{1},C_{3}are made equal. Hence the equation (4) reduces to R_{2}/R_{4}= 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 C

_{1}and C_{3}are fixed values. R_{1}and R_{3}are variable resistors controlled by a common shaft. Now providing R_{2}= 2R_{4}, 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 R

_{1}and R_{2}. Resistance R3 is connected in the standard arm. The fourth arm consists of an inductance L_{x}, capacitance C_{x}and resistance R_{x}.
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:

R

_{1}(R_{x}+ jωL_{x}– j/ωC_{x}) = R_{2}R_{3}
At resonance

X

_{L}= X_{c}and f_{x}= 1/2π√ (LC)
Z

_{x}= R_{x}
Therefore, R

_{x}= R_{2}R_{3}/R_{1}
As can be seen from the above the bridge is balanced by resistance alone. Resistance R

_{3}is used for this purpose.
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