Question

For a stable closed loop system, the gain at phase crossover frequency should always be:

• A. < 20 dB
• B. < 6 dB
• C. > 0 dB
• D. > 6 dB

Hint:

Always remember: For closed-loop stability, ensure the gain margin is positive at the phase crossover frequency — this translates to gain < 0 dB at phase −180◦ or equiva- lently, gain margin > 0 dB.

Answer 1

The Correct Option is C

In frequency response analysis of control systems, stability is often analyzed using Bode plots and the concept of gain and phase margins.

The phase crossover frequency is defined as the frequency at which the phase angle of the open-loop transfer function is \( -180^\circ \). At this frequency, if the magnitude of the open-loop gain is greater than 1 (or 0 dB), the system may oscillate and become unstable due to positive feedback.

However, for a closed-loop system to be stable, we must ensure that the gain at this frequency is less than 0 dB, or more precisely, the reciprocal condition in decibels implies that the gain margin should be greater than 0 dB.

This is a fundamental result from Nyquist and Bode stability criteria. It ensures that the system has a sufficient buffer against oscillations or instability caused by phase lag and gain increases.

Let’s review the options:
- Option (1): < 20 dB — Too vague and not directly relevant
- Option (2): < 6 dB — Again, arbitrary and not generalizable
- Option (3): > 0 dB — Correct, ensures positive gain margin and stability
- Option (4): > 6 dB — May be sufficient but not a necessary condition

Hence, the correct and general condition is that the gain margin at the phase crossover frequency should be greater than 0 dB.