Series RLC step/impulse behavior — match resistance conditions to the qualitative response (assume the standard form R compared to 2sqrt(L/C)). List I (Condition) A. R = 0 B. R 2sqrt(L/C) List II (Response) 1.

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Solve this multiple-choice question and choose the correct option.

Curioustab · Accepted Answer

Introduction:The damping nature of a series RLC circuit depends on the relative magnitude of R to the critical value 2sqrt(L/C). Recognizing underdamped, critically damped, overdamped, and ideal undamped cases is essential for filter, timing, and surge design. Given Data / Assumptions: Standard series RLC with resistance R, inductance L, capacitance C. Natural frequency ω0 = 1/sqrt(LC). Critical damping threshold at R = 2sqrt(L/C) (in SI units for series form). Concept / Approach: The characteristic equation roots determine the impulse/step response: underdamped (complex conjugate), critically damped (repeated real), overdamped (distinct real), and undamped (pure imaginary when R=0). Matching each region of R to its qualitative response follows directly from the discriminant of the second-order system. Step-by-Step Solution: R = 0 → zero damping → sustained sinusoidal oscillation → 1.R < 2sqrt(L/C) → complex roots → decaying oscillations → 2.R = 2sqrt(L/C) → repeated real root → fastest non-oscillatory response → 3.R > 2sqrt(L/C) → two negative real roots → slow non-oscillatory decay → 4. Verification / Alternative check: Compute damping ratio ζ = R / (2sqrt(L/C)). Then ζ < 1 underdamped, ζ = 1 critical, ζ > 1 overdamped; ζ = 0 gives undamped oscillation. Why Other Options Are Wrong: Swapping 2 and 4 confuses underdamped with overdamped behavior. Assigning A ≠ 1 implies oscillation even with resistance, contradicting R = 0 only. Common Pitfalls: Mixing the series and parallel RLC criteria; always check which topology is assumed. Also ensure consistent SI units when using 2sqrt(L/C). Final Answer: A-1, B-2, C-3, D-4

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