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 ω_pc, the linear gain margin is:

GM = 1/|L(jω_pc)|

Step 1 — Scale the Open-Loop Gain

L_new(s) = 2L_old(s)

A pure positive gain multiplier changes magnitude but not phase, so the phase-crossover frequency is unchanged.

Step 2 — Recalculate the Gain Margin

GM_new = 1/[2|L_old(jω_pc)|]

GM_new = (1/2)GM_old

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 = +20log_10(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 20log_10k.

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.

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 ω_pc, the linear gain margin is:

GM = 1/|L(jω_pc)|

Step 1 — Scale the Open-Loop Gain

L_new(s) = 2L_old(s)

A pure positive gain multiplier changes magnitude but not phase, so the phase-crossover frequency is unchanged.

Step 2 — Recalculate the Gain Margin

GM_new = 1/[2|L_old(jω_pc)|]

GM_new = (1/2)GM_old

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 = +20log_10(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 20log_10k.

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.

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 ω_pc, the linear gain margin is:

GM = 1/|L(jω_pc)|

Step 1 — Scale the Open-Loop Gain

L_new(s) = 2L_old(s)

A pure positive gain multiplier changes magnitude but not phase, so the phase-crossover frequency is unchanged.

Step 2 — Recalculate the Gain Margin

GM_new = 1/[2|L_old(jω_pc)|]

GM_new = (1/2)GM_old

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 = +20log_10(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 20log_10k.

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.