If the gain of an open-loop system is doubled, then the gain margin
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Answer & Explanation
Correct AnswerB. Decreases by factor of 2
Quick Explanation
Linear gain margin is the reciprocal of the open-loop magnitude at the phase-crossover frequency: GM = 1/|L(jωpc)|. Doubling the loop gain doubles that magnitude, so GMnew = GMold/2. Thus it decreases by a factor of 2.
Formula / Key Relation
GM = 1/|L(jωpc)|
Lnew(s) = 2Lold(s)
GMnew = GMold/2
20log10(2) = 6.02 dB
Detailed Explanation
Correct Answer
B — Decreases by factor of 2
Doubling the present open-loop gain uses up half of the remaining multiplicative gain reserve, so the linear gain margin is halved.
What This Question Is Really Testing
Gain margin answers: “By what factor can the loop gain still be increased before reaching the critical −180° condition?” Because it is an available reserve, it varies inversely with the existing loop magnitude at the phase-crossover frequency.
Concept Foundation
At the phase-crossover frequency ωpc, the linear gain margin is:
GM = 1/|L(jωpc)|
Step 1 — Scale the Open-Loop Gain
Lnew(s) = 2Lold(s)
A pure positive gain multiplier changes magnitude but not phase, so the phase-crossover frequency is unchanged.
Step 2 — Recalculate the Gain Margin
GMnew = 1/[2|Lold(jωpc)|]
GMnew = (1/2)GMold
Step 3 — State the Result
The gain margin is halved; therefore it decreases by a factor of 2. Option B is correct.
Why This Method Works
Gain margin is inversely related to the loop magnitude at phase crossover. Increasing the current gain moves the system closer to the critical unity-magnitude condition, leaving less additional gain available.
Independent Verification / Cross-Check
Gain ×2 ⇒ magnitude shift = +20log10(2)
= +6.02 dB
Therefore GM(dB) decreases by 6.02 dB
A 6.02 dB reduction is exactly the dB equivalent of halving the linear gain-margin factor.
Concept Extension
Linear gain margin is a ratio.
Gain margin in dB is logarithmic.
Do not mix “factor of 2” with “2 dB.”
Exam Strategy
For a pure loop-gain multiplier k, remember: linear GM divides by k; in dB, subtract 20log10k.
Quick Trick
Loop gain ×k → linear gain margin ÷k.
Why Other Options Are Wrong
A reverses the inverse relationship. C incorrectly introduces a square factor. D both reverses the direction and uses the wrong factor.
Common Mistake
Assuming loop gain and gain margin are directly proportional. Gain margin is the remaining reserve, so it changes inversely.
Exam Tip
Gain ×2 shifts the Bode magnitude up by 6.02 dB and reduces gain margin by the same 6.02 dB.
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