Barkhausen amplitude criterion in practice: To sustain oscillations in a feedback oscillator (e.g., Wien-bridge), how should the loop gain be managed so that the closed-loop condition settles?

Introduction / Context:
Oscillation start-up and steady-state behavior differ. Designers initially allow loop gain slightly above unity to grow a sinusoid from noise, then a control mechanism trims the gain to exactly unity at the oscillation frequency to prevent runaway or decay. This question clarifies that steady oscillation requires settling the effective closed-loop condition to unity loop gain.

Given Data / Assumptions:

• Linear analysis around the oscillation frequency f0.
• Positive feedback provides the frequency-selective path.
• Automatic amplitude control (lamp/JFET/compressor) is available.

Concept / Approach:
The Barkhausen magnitude criterion states |Aβ| = 1 at f0 for sustained oscillation. Practical circuits start with |Aβ| > 1 to guarantee build-up, then introduce negative feedback or gain compression so that the effective loop gain is reduced to 1. If it remains > 1, the waveform clips; if < 1, oscillations die out. Therefore, “reduced, equals one” describes steady operation best.

Step-by-Step Solution:

Verification / Alternative check:
Classic lamp-stabilized Wien oscillators use the lamp’s resistance rise with temperature to decrease gain as amplitude increases, naturally converging on unity loop gain with low distortion. Similar behavior is achieved with JFET-based AGC or diode limiters with careful biasing.

Why Other Options Are Wrong:
“Reduced, less than one” kills oscillation; “increased, more/ much more than one” leads to runaway and clipped output; leaving gain uncontrolled invites severe distortion or saturation.

Common Pitfalls:
Thinking Barkhausen requires > 1 at all times; in fact, > 1 is only for start-up. Long-term, unity magnitude and 0° phase are required.

Final Answer:
Reduced, equals one