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 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:

R₁(R_x + jωL_x – j/ωC_x) = R₂ R₃

At resonance

X_L = X_c and f_x = 1/2π√ (LC)

Z_x = R_x

Therefore, R_x = R₂ R₃/R₁

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