Barkhausen criterion for oscillators: start-up versus steady state For a linear feedback oscillator to start and then sustain a stable sinusoidal oscillation, what condition must the loop gain satisfy at the oscillation frequency (magnitude and phase), and how should the amplifier gain be set conceptually?

Introduction / Context:
Oscillators rely on positive feedback to convert DC power into a sustained AC signal without an external periodic input. The Barkhausen criterion summarizes the steady-state condition required for sustained oscillation and guides practical gain setting for reliable startup and amplitude stabilization.

Given Data / Assumptions:

• Linear time-invariant amplifier with a frequency-selective feedback network B(jω).
• Focus at the intended oscillation frequency ω0.
• Small-signal linearization applies around the operating point; nonlinearities eventually limit amplitude.

Concept / Approach:
The Barkhausen criterion states that at ω0 the loop gain must satisfy two conditions: magnitude |Av * B| = 1 and net phase shift 0° (or 360°). Practically, to ensure startup from noise, designers choose |Av * B| slightly greater than 1. As amplitude grows, nonlinearities reduce effective gain until the loop gain settles to unity, sustaining a constant amplitude sinusoid.

Step-by-Step Solution:

Verification / Alternative check:
Simulations show noise seeds a small signal that grows when |Av * B| > 1. As amplitude increases, gain compression occurs and the loop gain returns to unity. Removing feedback or deviating phase from 0° kills or detunes oscillation.

Why Other Options Are Wrong:

Common Pitfalls:
Confusing start-up (slightly > 1) with steady-state (exactly 1); ignoring the phase requirement; assuming “more gain is always better,” which increases distortion.

Final Answer:
Set gain so |Av * B| = 1 with 0° net phase (≈ 360°); slight >1 at startup, then settles to 1