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?
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ASet gain so |Av * B| < 1; net phase ≈ 0°
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BSet gain so |Av * B| = 1 with 0° net phase (≈ 360°); slight >1 at startup, then settles to 1
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CSet gain so |Av * B| ≫ 1 at all times; any phase is acceptable
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DSet gain so |Av * B| = 0.5 with 180° net phase
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ESet gain so |Av * B| = 1 with 180° net phase
Answer
Correct Answer: Set gain so |Av * B| = 1 with 0° net phase (≈ 360°); slight >1 at startup, then settles to 1
Explanation
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