Wien-bridge oscillator working

A Wien-bridge oscillator is an audio-frequency RC oscillator which uses an op-amp (non-inverting amplifier) with a Wien-bridge network as the positive-feedback path. It gives a low-distortion sine wave.

Circuit diagram

              +-----[ Rf ]-----+
              |                |
        +-----+-----(-)\       |
        |                >-----+------o Vo
        |     +-----(+)/       |
       [R1]   |                |
        |     +---[R]---[C]----+   series arm: Vo to (+)
       GND    |
              +---[R]--+
              |        |
              +---[C]--+
              |
             GND                   parallel arm: (+) to ground

The series arm (R in series with C) joins Vo to the (+) input, and the parallel arm (R in parallel with C) joins the (+) input to ground. Rf and R1 form the gain-setting divider at the (−) input.

• Positive feedback: series R-C from the output to the (+) input, and parallel R-C from the (+) input to ground (the Wien network).
• Negative feedback: the divider Rf and R1 (sets the closed-loop gain).

Working and expressions

The feedback factor of the Wien network (with equal R and C) is

β = Vf/Vo = 1 / [3 + j(ωRC − 1/(ωRC))]

• At ω = ωo = 1/(RC): the reactive part is zero, so β = 1/3 and the phase shift is 0°.
• Barkhausen condition: the loop gain Aβ must be unity with 0° phase, so the amplifier gain must be

A = 1 + Rf/R1 = 3 → Rf = 2 R1

• Frequency of oscillation:

fo = 1 / (2π R C)

(If the arm values are unequal, with Ra and Ca in the series arm and Rb and Cb in the parallel arm: fo = 1/(2π √(Ra Rb Ca Cb)).)

To start the oscillation, the gain is made slightly above 3 (Rf slightly more than 2R1), and the amplitude is limited by a nonlinear element (a lamp or a diode network in the feedback). Example: R = 10 kΩ, C = 16 nF gives fo ≈ 1 kHz.