In the series RLC circuit in the figure, R = 5 Ω, L = 1 H and C = 0.25 F. Which pair correctly gives the roots of the characteristic equation and the type of step response?

Answer: C. s = −1 and s = −4, overdamped

Method 1 (characteristic equation): substituting the values into Ls² + Rs + 1/C = 0 gives s² + 5s + 4 = 0, and (s + 1)(s + 4) = 0 gives two distinct negative real roots, s = −1 and −4. The response is a sum of terms in e^(−t) and e^(−4t) and approaches the final value without oscillating: it is overdamped. Method 2 (damping ratio): ωn = 1/√(LC) = 2 rad/s and ζ = (R/2)√(C/L) = 2.5 × 0.5 = 1.25; since ζ > 1, the response is overdamped. The discriminant R² − 4L/C = 25 − 16 = 9 > 0 confirms this.

• A. −2 is the value of ωn = 1/√(LC). Critical damping (ζ = 1) occurs when R = 2√(L/C) = 4 Ω.
• B. This attaches ωn = 2 directly as the imaginary part to the real part −R/(2L) = −2.5. The imaginary part is √(ωn² − α²), and here ωn < α, so there is no imaginary part.
• C. Correct. s = −1 and −4, overdamped.