A Wien Bridge is a bridge circuit used for measuring an unknown capacitance by balancing the loads of its four arms, one of which contains the unknown capacitance. Figure 1 below shows a diagram of the Wien Bridge.  

Figure 1.  The Wien Bridge

As shown in Figure 1, one arm of a Wien bridge consists of a capacitor in series with a resistor (C1 and R3) and another arm consists of a capacitor in parallel to a resistor (C2 and R4).  The other two arms simply contain a resistor each (R1 and R2).  The values of R1and R2 are known, and R4 and C2 are both adjustable. The unknown values are those of C1 and R3.

Like other bridge circuits, the measuring ability of a Wien Bridge depends on 'balancing' the circuit. Balancing the circuit in Figure 1 means adjusting R4 and C2 until the current through the ammeter between points A and B becomes zero.  This happens when the voltages at points A and B are equal.  When the Wien Bridge is balanced, it follows that R2/R1 = Z1/Z2 where Z1 is the impedance of the arm containing C1 and Z2 is the impedance of the arm containing C2. 

Mathematically, when the bridge is balanced,

R2/R1 = (1/ωC1 + R3) / (R4/[ωC2(R4 + 1/ωC2)]) wherein ω = 2πf; or

R2/R1 = (1/ωC1 + R3) / (R4/[ωC2R4 + 1]); or

R2/R1 = (1/ωC1 + R3) (ωC2 + 1/R4); or

R2/R1 = C2/C1 + ωC2R3 + 1/(ωC1R4) + R3/R4.

When the bridge is balanced, the capacitive reactances cancel each other out, so 

R2/R1 = C2/C1 + R3/R4. Thus, C2/C1 = R2/R1 - R3/R4.

Note that the balancing of a Wien Bridge is frequency-dependent.  The frequency f at which the Wien Bridge in Figure 1 becomes balanced is the frequency at which ωC2R3 = 1/(ωC1R4), or 2πfC2R3 = 1/(2πfC1R4). Thus, the frequency f is given by the following equation:  f = (1 / 2π) x (sqrt(1/[R3R4C1C2])).

See Also:   Bridge Circuits;  More Articles