Bode stability insight: According to the Bode stability criterion, a linear time-invariant feedback system is unstable if the open-loop amplitude ratio exceeds unity at the frequency where the phase lag equals:

Introduction / Context:The Bode stability criterion offers a frequency-domain way to infer closed-loop stability from open-loop frequency response data. It links gain and phase at critical frequencies to predict whether a system will oscillate or diverge after closing the loop.

Given Data / Assumptions:

• Single-loop, negative-feedback, linear time-invariant system.
• Open-loop transfer L(jω) measured or modeled.
• Amplitude ratio refers to |L(jω)| and phase lag to ∠L(jω) (negative angle convention).

Concept / Approach:The Bode criterion states that if, at the phase crossover (where the phase lag reaches 180°, i.e., ∠L = -180°), the magnitude |L| exceeds 1 (0 dB), the Nyquist plot encircles the -1 point and the closed loop becomes unstable. Equivalently, at the gain crossover (|L| = 1), sufficient negative phase margin (distance from -180°) is required to maintain stability.

Step-by-Step Solution:

Verification / Alternative check:Relating Bode and Nyquist: at -180° phase, a gain above unity places the Nyquist plot beyond the -1 point, implying an encirclement upon closure (instability).

Why Other Options Are Wrong:

• 0°, 45°, 90°: These lags are not the Nyquist critical angle for sign reversal in feedback.
• 270°: Not the conventional critical lag for single-loop negative feedback using Bode margins.

Common Pitfalls:Confusing gain crossover and phase crossover; the instability check at phase crossover uses the magnitude condition, and at gain crossover uses the phase margin condition.

Final Answer:180°