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.

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Solve this multiple-choice question and choose the correct option.

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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: Ensure the frequency-selective network provides a 0° net phase at ω0 (e.g., RC phase shift totals 180° with inverting amplifier phase −180°, etc.).Choose amplifier gain Av so that |Av * B| > 1 at cold start (e.g., 1.05–1.5 depending on design).Allow device or AGC/nonlinear limiting to reduce Av as the oscillation builds until |Av * B| → 1.At steady state, oscillation is sustained with |Av * B| = 1 and net phase 0°. 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: |Av * B| < 1: loop attenuates; no sustained oscillation.|Av * B| ≫ 1 always: produces distortion and clipping; steady state still forces effective loop gain to 1 via nonlinearity.180° phase or 0.5 magnitude: represent negative feedback or insufficient loop gain. 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

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