Which of the following is closest to the gain margin of the loop transfer function L(s) = 2/(s(s + 1)(s + 2))?

Answer: B. About 9.5 dB

The phase is −90° − tan⁻¹ω − tan⁻¹(ω/2); the two angles add up to 90° when ω × (ω/2) = 1, so the phase reaches −180° at ω = √2 rad/s. The gain there is |L(j√2)| = 2/(√2 × √3 × √6) = 1/3, so the gain margin is 20 log₁₀3 ≈ 9.5 dB. Multiplying the gain by 3 (K = 6) would reach the limit, which agrees with the stable range 0 < K < 6 found with the Routh–Hurwitz stability criterion.

• A. This is 10 log₁₀3. Ratios of voltages or gains are converted to decibels with 20 log₁₀.
• B. Correct. There is a factor of 3 of margin before the critical gain.
• C. This expresses the critical gain 6 directly in decibels. The margin is its ratio to the present gain 2, which is 3.