In the series RLC circuit in the figure, L = 1 mH, C = 0.1 µF and R = 50 Ω. Which pair correctly gives the damping ratio ζ and the type of step response?

Answer: A. ζ = 0.25, damped oscillation

Method 1 (compare with the standard form): the characteristic equation is s² + (R/L)s + 1/(LC) = 0, and comparing it with the standard form s² + 2ζωn s + ωn² gives ωn = 1/√(LC) = 1×10⁵ rad/s and 2ζωn = R/L. From these, ζ = (R/2)√(C/L) = 25 × √(0.1×10⁻⁶/1×10⁻³) = 25 × 0.01 = 0.25. Method 2 (ratio to the critical resistance): the resistance for critical damping is 2√(L/C) = 200 Ω, so ζ = R/(2√(L/C)) = 50/200 = 0.25 as well. Since ζ < 1, the step response overshoots the final value and settles while oscillating: a damped oscillation.

• A. Correct. ζ = 50/200 = 0.25, a damped oscillation.
• B. R√(C/L) = 0.5 forgets the factor 1/2 in the formula for ζ.
• C. 4.0 is the reciprocal of ζ, mixing up √(L/C) and √(C/L).