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Wien-Bridge Oscillator

Active analog

A non-inverting op-amp wrapped in a Wien RCRC network. When the loop gain crosses unity at exactly one frequency, thermal noise is amplified into a clean sinusoid; a nonlinear gain limiter parks the amplitude on a stable limit cycle. The phase portrait shows the oscillation being born — a Hopf bifurcation you can trigger with a knob.

12 components · 15 connections · second-order loop with amplitude limiting

the aha

Feed an amplifier a whisper of its own output — in phase, at one special frequency — and it will build a pure tone from nothing but noise.

why it works
The Barkhausen criterion is deceptively simple: for sustained oscillation the signal must travel around the loop and come back exactly in phase and exactly the same size, i.e. loop gain Aβ=1A\beta = 1. The Wien network passes only one frequency f0=12πRCf_0 = \frac{1}{2\pi RC} with zero phase shift and a 13\frac{1}{3} attenuation, so the amplifier must supply a gain of exactly 33 to close the loop there and nowhere else. Set the gain a hair above 33 and any noise at f0f_0 grows exponentially; a gentle nonlinearity then pulls it back, and the amplitude settles onto a limit cycle. Cross the gain through 33 and you have crossed a Hopf bifurcation — a fixed point becoming a stable oscillation, drawn live in the phase portrait.
governing model
f0=12πRC  ⋅  Aβ=1 (Barkhausen), gain =3f_0 = \dfrac{1}{2\pi RC} \;\cdot\; A\beta = 1 \text{ (Barkhausen), gain } = 3
structural inventory
  • op-amp gain stage
  • series + shunt RCRC bridge
  • Rf/RgR_f/R_g gain setting
  • amplitude-limiting element
boundaries
  • gain 2.82.8–3.23.2
  • f0f_0 from RCRC
  • Hopf onset at gain =3=3
fault relay gain-set resistor open (latch-up)

  1. Raise the gain from below 3.

    Under 33, disturbances decay — the output dies to silence. That is a stable fixed point.

  2. Cross gain = 3 and watch the tone ignite.

    Just past 33, the phase portrait blooms from a point into a closed loop: oscillation born from noise. This birth-of-a-cycle is the Hopf bifurcation.

  3. Open the gain-set resistor.

    Gain runs away, the op-amp slams into its rails, and the pure tone becomes a clipped square — a real failure mode of under-designed oscillators.
causal signal trace paused

Amplify a perturbation Noise at the op-amp input is the seed; loop gain decides whether it grows.

step 1 / 5 Amplify a perturbation

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