Gain Margin and Phase Margin of an Unstable System

Gain margin and phase margin are two important parameters used to analyze the stability and robustness of a control system. These margins indicate the system's ability to tolerate variations in gain and phase before becoming unstable.

1. Positive Gain Margin:
A positive gain margin indicates that the system's gain is below unity at the frequency where the phase angle reaches -180 degrees. In other words, the system has some additional gain capacity before it becomes unstable. This means that the system can tolerate a certain amount of gain variation without losing stability.

2. Negative Gain Margin:
A negative gain margin implies that the system's gain is already above unity at the frequency where the phase angle reaches -180 degrees. This indicates that the system is on the verge of instability and any further increase in gain can cause the system to become unstable.

3. Positive Phase Margin:
A positive phase margin indicates that the system's phase angle is greater than -180 degrees at the frequency where the gain is unity. This means that the system can tolerate a certain amount of phase lag before becoming unstable. A larger phase margin implies a more robust and stable system.

4. Negative Phase Margin:
A negative phase margin implies that the system's phase angle is already less than -180 degrees at the frequency where the gain is unity. This indicates that the system is prone to instability and any further phase lag can cause the system to become unstable.

Explanation of Correct Answer:

Option D states that statements 2, 3, and 4 are correct. Let's analyze each statement:

2. Positive gain margin and negative phase margin:
This statement is correct. A positive gain margin indicates that the system can tolerate some gain variation before becoming unstable. However, a negative phase margin implies that the system is already close to instability due to excessive phase lag.

3. Negative gain margin and positive phase margin:
This statement is correct. A negative gain margin indicates that the system is on the verge of instability due to excessive gain. On the other hand, a positive phase margin implies that the system can tolerate some phase lag before becoming unstable.

4. Negative gain margin and negative phase margin:
This statement is also correct. Both a negative gain margin and a negative phase margin indicate that the system is already close to instability in terms of both gain and phase.

Therefore, option D correctly identifies that statements 2, 3, and 4 are correct.