For reliable start-up of a Wien-bridge sine-wave oscillator, what should the initial closed-loop gain A_cl of the amplifier be relative to the ideal value of 3 required by the feedback network?

Introduction:
The Wien-bridge oscillator is prized for low-distortion sine waves. It relies on a frequency-selective feedback network that requires a loop gain slightly above unity at start-up to initiate oscillation, then near-exactly unity to sustain a constant amplitude. Understanding how A_cl relates to the required loop gain is key to practical design.

Given Data / Assumptions:

• Wien bridge RC network demands an amplifier gain of about 3 for unity loop gain at the oscillation frequency.
• Start-up must overcome losses and component tolerances.
• Amplitude stabilization (e.g., lamp, JFET, diodes) will later reduce gain toward the ideal value.

Concept / Approach:
To start oscillation, the Barkhausen magnitude condition must be slightly exceeded (loop gain > 1). In a Wien-bridge, that equates to setting A_cl a bit greater than 3 initially. A stabilization element then automatically lowers A_cl toward 3 as the output grows, maintaining steady amplitude with low distortion.

Step-by-Step Solution:
Set initial A_cl slightly > 3 to ensure loop gain > 1.As the output amplitude rises, a nonlinear element (lamp resistance increase, JFET resistance change, diode limiter) reduces effective gain.The loop gain converges to 1 at steady state (A_cl ≈ 3).Result: stable sinusoidal oscillation at the design frequency.

Verification / Alternative check:
Classic Hewlett-Packard designs used a small incandescent lamp in the feedback path; its resistance rises with temperature, automatically trimming the gain from >3 at start-up to ≈3 during steady operation—experimental proof of the principle.

Why Other Options Are Wrong:

• A_cl < 3 or ≅ 1: Loop gain below unity prevents start-up.
• 0: No amplification, no oscillation.
• 3/2: Far too low to start.

Common Pitfalls:

• Setting A_cl exactly 3 without any start-up margin; unit tolerances can suppress oscillation.
• Excessive initial gain causing heavy clipping and distortion if stabilization is too slow.

Final Answer:
A_cl > 3