A system has gain crossover frequency at which magnitude is unity and phase is –150 degrees, then1.The phase margin of the system is 30 degrees.2.The system is stable based on phase margin criterion.3.Increasing gain may lead to instability.4.Gain margin is zero at this operating point.Which of the above statements is/are correct?

Correct Answer: D - 1, 2 and 3 only

Quick Concept Explanation

Gain margin and phase margin quantify distance from the critical feedback condition and are widely used as relative-stability measures; pole and zero locations shape the frequency response. At gain crossover, phase=−150°, so PM=180−150=30°. Gain margin must be evaluated at phase crossover, so it is not automatically zero here.

Exam focus: Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions. Common trap: Frequency response is not independent of poles and zeros.

Statement-wise Verification

• Statement 1 - Correct.
  The phase margin of the system is 30 degrees.
  At gain crossover, phase=−150°, so PM=180−150=30°.
• Statement 2 - Correct.
  The system is stable based on phase margin criterion.
  Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions.
• Statement 3 - Correct.
  Increasing gain may lead to instability.
  At gain crossover ω_gc: PM=180°+∠G(jω_gc).
• Statement 4 - Incorrect.
  Gain margin is zero at this operating point.
  Gain margin is evaluated at phase-crossover frequency, not inferred as zero merely because magnitude is unity at gain crossover

Core Concept

Frequency response describes how a linear system's gain and phase vary with sinusoidal frequency. Bode plots display logarithmic magnitude and phase. Gain margin and phase margin quantify distance from the critical feedback condition and are widely used as relative-stability measures; pole and zero locations shape the frequency response.

Formula / Key Relationship

At gain crossover ω_gc: PM=180°+∠G(jω_gc).

Step-by-Step Check

At gain crossover, phase=−150°, so PM=180−150=30°. Gain margin must be evaluated at phase crossover, so it is not automatically zero here.

Why the Other Options Are Wrong

Option A is incorrect because it includes incorrect statement(s) 4. Option B is incorrect because it includes incorrect statement(s) 4 and omits correct statement(s) 2, 3. Option C is incorrect because it omits correct statement(s) 1.

Exam Shortcut / Approach

Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions.

Common Mistake

Frequency response is not independent of poles and zeros.

Quick Revision

Phase Margin and Gain Crossover is linked with Stability Margins, Frequency Response and Control Systems. Positive phase margin usually indicates a stable negative-feedback loop with normal assumptions.