In the series RLC circuit in the figure, L = 1 mH and C = 0.1 µF. Which of the following is the resistance R that gives critical damping, in which the step response approaches the final value fastest without oscillating?

Answer: B. 200 Ω

Method 1 (characteristic equation): from the circuit equation L di/dt + Ri + (1/C)∫i dt = E, the characteristic equation is Ls² + Rs + 1/C = 0. The condition for a double root (zero discriminant) is R² = 4L/C, so R = 2√(L/C) = 2 × √(1×10⁻³/0.1×10⁻⁶) = 2 × 100 = 200 Ω. Method 2 (damping ratio): comparing with the standard form s² + 2ζωn s + ωn² gives ζ = (R/2)√(C/L), and critical damping ζ = 1 gives the same R = 2√(L/C) = 200 Ω. A smaller R gives a damped oscillation, and a larger R gives overdamping.

• A. √(L/C) = 100 Ω forgets the factor of 2. This value gives a damped oscillation with ζ = 0.5.
• B. Correct. R = 2√(L/C) = 200 Ω.
• C. This forgets to take the square root of L/C = 10⁴ (the unit would also become Ω²).