Introduction / Context:
Oscillators generate periodic waveforms without an external input signal by satisfying specific feedback conditions. The Barkhausen criterion captures the essence of these conditions in terms of loop gain magnitude and phase. This question asks you to identify the required total phase shift around the loop for sustained oscillation.
Given Data / Assumptions:
- Linear time-invariant small-signal model near the intended oscillation frequency.
- Loop gain L(jω) = A(jω) * β(jω), where A is the amplifier transfer and β is the feedback network transfer.
- Frequency is such that the oscillator is designed to start and sustain oscillations.
Concept / Approach:
The Barkhausen criterion states two key conditions at the oscillation frequency ω0: (1) magnitude condition |L(jω0)| = 1 and (2) phase condition ∠L(jω0) = 0° modulo 360°. The '0° modulo 360°' requirement ensures that the fed-back signal reinforces (is in phase with) the original, enabling self-excitation without an external drive.
Step-by-Step Solution:
Write L(jω) = A(jω) β(jω). For reinforcement, the returning signal must have the same phase as the original.Hence the total loop phase shift ∠A + ∠β must equal 0° + 360°k (k is an integer). The smallest positive equivalent is 360°.Simultaneously, |Aβ| must equal 1 at ω0 to avoid decay (<1) or divergence (>1).
Verification / Alternative check:
Common RC phase-shift oscillators produce 180° in the RC network and 180° in the inverting amplifier, totaling 360°.
Why Other Options Are Wrong:
90°, 45°: These do not produce constructive reinforcement around the loop.270°: Often seen in part of the loop, but the total must be 0° mod 360°.180°: Alone implies inversion; without an additional 180° from elsewhere, feedback is negative, not regenerative.
Common Pitfalls:
Assuming 180° is sufficient; forgetting that the net phase after one loop must be an integer multiple of 360°.
Final Answer:
360°
Discussion & Comments