Which of the following condition should be satisfied for ensuring stability of an LTI system?
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Answer & Explanation
Correct AnswerD. Both phase margin and gain margin should be positive.
Quick Explanation
In the standard negative-feedback frequency-response stability test used in this question, both margins must be positive: GM > 0 dB and PM > 0°. Positive margins mean the loop has not yet reached the critical instability boundary.
Formula / Key Relation
Gain Margin: GM > 0 dB
Phase Margin: PM > 0°
Detailed Explanation
Correct Answer
D — Both phase margin and gain margin should be positive.
Positive gain margin and positive phase margin place the nominal loop on the stable side of the standard frequency-response stability boundary.
What This Question Is Really Testing
This question tests what gain margin and phase margin mean physically. They are not the gain and phase themselves; they are distances from the critical condition that would cause sustained oscillation in the standard negative-feedback setting.
Concept Foundation
Gain margin (GM) is evaluated at the phase-crossover frequency where the open-loop phase is −180°. Phase margin (PM) is evaluated at the gain-crossover frequency where the open-loop magnitude is unity.
Step 1 — Check Gain Margin
At the phase-crossover frequency, the loop magnitude should still be below unity so that some additional gain can be tolerated.
GM > 0 dB
Step 2 — Check Phase Margin
At the gain-crossover frequency, the phase should still be more positive than −180°, leaving a positive phase reserve.
PM > 0°
Step 3 — Apply the Exam Criterion
Both required margins are positive, so the correct choice is D.
Why This Method Works
At the critical oscillatory boundary, gain margin or phase margin becomes zero. Positive values indicate that the loop still has reserve before that boundary is reached.
Independent Verification / Cross-Check
This is the standard Bode/Nyquist margin interpretation used in basic control-system MCQs. Exact stability for unusual/nonminimum-phase cases should always be judged from the full Nyquist/pole information, but that is beyond what this item is testing.
Concept Extension
GM is commonly expressed in dB.
PM is expressed in degrees.
Larger positive margins usually mean greater relative stability, though they also affect bandwidth and response speed.
Exam Strategy
Do not confuse the phase value −180° with phase margin. For this class of MCQ, remember: stable side → GM positive and PM positive.
Quick Trick
Both margins positive → standard stable answer.
Why Other Options Are Wrong
A has both margins negative. B has negative gain margin. C has negative phase margin. Each places at least one margin on the wrong side of the standard stability criterion.
Common Mistake
Thinking that because the phase-crossover angle is −180°, the phase margin should also be negative. Phase margin is the positive angular reserve to −180°.
Exam Tip
Memorize the standard margin check: GM > 0 dB and PM > 0°.
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