Wien-bridge oscillator condition: For a Wien-bridge oscillator to produce a sustained sine-wave output, must the net phase shift around the positive-feedback loop be greater than 0° (i.e., nonzero)?

Introduction / Context:The Wien-bridge oscillator is a classic RC sine-wave source built around an operational amplifier with a frequency-selective feedback network. Its steady, low-distortion oscillation hinges on satisfying the Barkhausen criteria: a loop gain of unity at the oscillation frequency and a net loop phase shift of 0° (or an integer multiple of 360°). This item probes whether the required loop phase must be greater than 0°, which would contradict the fundamental condition for sustained oscillation.

Given Data / Assumptions:

• Single-op-amp Wien-bridge topology with an RC lead-lag network in the positive feedback path.
• Linear, small-signal analysis at the intended oscillation frequency f0.
• Amplitude control (lamp/JFET/AGC) ensures loop gain settles to unity but does not alter the phase criterion.

Concept / Approach:The Barkhausen phase condition requires the total phase shift around the closed loop to be 0° (mod 360°) at f0 so that the fed-back signal reinforces the input. In the Wien network, the RC bridge contributes 0° phase shift exactly at f0, while at other frequencies it leads or lags. The op-amp stage is arranged to provide non-inverting amplification so the overall phase is 0° at f0. Any net phase greater than 0° (i.e., a non-zero residual phase at f0) would reduce constructive reinforcement and prevent a steady, sustained sinusoid.

Step-by-Step Solution:

Verification / Alternative check:Frequency response of the Wien network shows a flat phase of 0° at its center frequency, with positive/negative phase away from f0. Practical oscillators add an amplitude-control element (e.g., incandescent lamp or JFET) to trim loop gain to unity while preserving the 0° phase at f0.

Why Other Options Are Wrong:“Only during start-up” misattributes phase; start-up requires loop gain > 1, not phase > 0°. LC tanks are not part of the Wien bridge. Transistor type is irrelevant to the phase condition at f0.

Common Pitfalls:Confusing sign convention: “greater than 0°” would imply a nonzero phase error, violating Barkhausen. Also mixing amplitude criterion (gain control) with the independent phase criterion.

Final Answer:Incorrect