Series RLC circuit – condition for underdamped oscillations For a series RLC circuit driven away from steady-state, under what condition on R, L, and C will the natural response be underdamped (i.e., oscillatory with an exponentially decaying envelope)?

Introduction / Context:
In circuit theory and power electronics, the transient response of a series RLC (resistor–inductor–capacitor) network can be underdamped, critically damped, or overdamped. Recognizing the mathematical condition for each regime is essential for designing commutation networks, filters, snubbers, and resonant converters.

Given Data / Assumptions:

• Series RLC with elements R, L, and C.
• Natural (homogeneous) response considered.
• Linear, time-invariant components; no non-ideal losses beyond R.

Concept / Approach:

The differential equation for a series RLC (current i as the state) leads to the characteristic equation s^2 + (R/L)s + 1/(LC) = 0. The damping depends on the discriminant Δ = (R/L)^2 − 4/(LC). If Δ < 0, complex conjugate roots occur and the response is underdamped (oscillatory). If Δ = 0, it is critically damped; if Δ > 0, it is overdamped.

Step-by-Step Solution:

Verification / Alternative check:

Define damping factor α = R/(2L) and undamped natural frequency ω0 = 1/sqrt(LC). Underdamped occurs when α < ω0, i.e., R/(2L) < 1/sqrt(LC) → R^2 < 4L/C, which matches the condition derived above.

Why Other Options Are Wrong:

• R^2 = 4L/C (option b): This is the critically damped boundary, not underdamped.
• R^2 > 4L/C (option c): Overdamped, producing non-oscillatory exponentials.
• R = 0 only (option d): R = 0 yields purely sinusoidal (undamped) oscillation, not the general underdamped case with decay.

Common Pitfalls:

Confusing the undamped (R = 0) case with underdamped; underdamped requires finite R but less than the critical value. Another mistake is mixing series and parallel RLC conditions; their formulas differ.

Final Answer:

R^2 < 4L/C