Wien Bridge Oscillator: Nyquist Theory and Oscillation Startup

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This accessible text edition preserves the document's narrative. See the interactive edition for typeset equations, diagrams and playback.

A Circuit that Oscillates

A non-inverting op-amp drives a Wien RC bridge. The series RC arm joins the output to the positive input; the parallel RC arm joins that input to ground. Local negative feedback through Rf and Rg sets amplifier gain A=1+Rf/Rg. The diagram is a functional circuit: supply rails, bias, device bandwidth and practical limiter implementation are omitted. It is not a validated construction schematic. With equal R=10 kilohms and C=10 nanofarads: _0=1RC=10000\,rad/s, f_0=12 RC1591.55\,Hz. At this frequency the bridge introduces zero phase shift and attenuates the signal to one third.

Positive Feedback and the Nyquist Critical Point

(s)=sRC(sRC)^2+3sRC+1, L(s)=A(s), T(s)=A1-A(s). The characteristic equation is 1-L(s)=0. Therefore this positive-feedback convention uses critical point +1, not -1. Equivalently define Ln=-L and use the familiar negative-feedback equation 1+Ln=0 with critical point -1. Do not mix the two sign conventions. At q=omega RC: L(j)=Ajq1-q^2+j3q, L(j_0)=A3. The Nyquist locus passes through +1 when A=3. The characteristic polynomial in p=sRC is: p^2+(3-A)p+1=0. For A below 3 the poles lie in the left half-plane and a disturbance decays. At A=3 the ideal linear poles are plus and minus j, giving neutral oscillations whose amplitude depends on initial conditions. Above 3 a right-half-plane pole pair causes small perturbations to grow. This is the startup condition, not a stable bounded output by itself. For this proper plant there are no imaginary-axis open-loop poles; the positive and negative frequency curves approach the origin at both extremes.

Why a Real Oscillator Needs Amplitude Control

Barkhausen's unity loop gain and zero phase condition describes the sinusoidal steady-state boundary; it is not by itself sufficient to prove startup, global stability or a unique oscillation amplitude. Noise or a small initial perturbation seeds startup when A exceeds 3. As amplitude grows, a lamp, diodes, AGC or other nonlinear mechanism reduces effective loop gain. Op-amp rail clipping alone produces distortion rather than a clean controlled sine wave.

Animated Time-Domain Model

The animation uses normalized time tau=omega0 t and a dimensionless output x: d^2xd^2+(3-A+0.8x^2)dxd+x=0. This educational nonlinear damping model has the same small-signal characteristic polynomial as the Wien oscillator. It illustrates startup and amplitude limiting but is not a device-level simulation of a lamp, diode circuit or op-amp. At A=3.3 negative small-signal damping grows the seed, while the nonlinear term limits the amplitude. At A=3 nonlinear damping slightly reduces a finite seed; neutral sustained oscillation is the linearized boundary, not a claim about this nonlinear trace. Initial conditions are x=0.02 and dx/dtau=0. The solver uses fourth-order Runge-Kutta, timestep 0.003, and no external periodic drive.

Animated Conventional Current

Moving dots follow conventional current, reversing when the signed branch current reverses. Green denotes the reference direction and orange the opposite direction; dot size indicates relative magnitude. The ideal op-amp input draws no current, so the positive-input wire carries no animated flow. No dots cross a capacitor dielectric; lead currents depict capacitor charging and discharging. For normalized output x, bridge voltage b and series-capacitor voltage c, the passive equal-RC bridge is driven by the simulated output: i_s=x-c-b, i_R=b, i_C=i_s-i_R, dbd=i_C, dcd=i_s. Branch values are normalized, not amperes. The passive bridge uses forward Euler with timestep 0.003. It is a driven readout of the educational oscillator waveform, not a full nonlinear circuit solution. The local feedback current assumes Rg=R and Rf=(A-1)R. Supply and output-stage currents are not shown. Capacitor energy rises when b times iC is positive and falls when it is negative.

Interactive Circuit, Nyquist Locus and Waveform

Play sweeps the Nyquist frequency logarithmically while revealing the time waveform. These share a teaching timeline but represent different independent variables: frequency sweep is not physical oscillator time. Select Decay, Boundary or Startup and replay to compare behavior. The waveform spans 72 normalized time units, approximately 7.2 milliseconds of physical time, slowed to 24 seconds of playback. Output amplitude is dimensionless, not a calibrated voltage.