A power supply output filter must suppress switching transients without any voltage oscillation on the output rail. Which damping regime is most appropriate, and why?
AUnderdamped (ζ < 1), because it settles to the final value fastest
BCritically damped (ζ = 1), because it is always the optimal response
COverdamped (ζ > 1), because transients decay monotonically without any overshoot or oscillation
DAny damping regime works; the choice does not affect output quality
A power supply output must not oscillate — voltage ringing on the supply rail would corrupt downstream circuits. Overdamping ensures the transient decays smoothly and monotonically, never crossing the final value. Underdamped response would ring, and critically damped, while free of overshoot, sits exactly at the boundary and can still ring due to component tolerances. The cost of overdamping is slower settling, which is acceptable for a supply filter but not for other applications.
Question 2 Multiple Choice
A student claims: 'Critical damping is always the best RLC response because it's the fastest without any overshoot.' Which scenario best refutes this claim?
AA high-Q radio receiver tank circuit, which needs strong underdamped resonance to select one frequency and reject adjacent ones
BA motor controller that needs to avoid overshoot to prevent mechanical damage
CA filter designed to reject high-frequency noise on a power supply
DA digital circuit that requires clean voltage transitions without ringing
A radio receiver tank circuit is intentionally underdamped (ζ ≪ 1, very high Q). The strong ringing at a specific frequency is the desired behavior — it allows the circuit to resonate at the target radio frequency and reject nearby frequencies. Critical damping, which suppresses all oscillation, would completely destroy this selectivity. 'Best' is always application-dependent: critical damping is fastest non-oscillatory; underdamped is preferred when resonance or faster-to-threshold settling (accepting overshoot) is desired; overdamped is preferred when monotonic decay is required.
Question 3 True / False
An RLC circuit with zero resistance (ζ = 0) will oscillate at frequency ω_n indefinitely without any amplitude decay.
TTrue
FFalse
Answer: True
With no resistance, there is no mechanism to dissipate energy. The circuit has purely imaginary poles at s = ±jω_n, which in the time domain corresponds to a pure sinusoid with constant amplitude — sustained oscillation at the natural frequency. Mathematically, the step response includes a term sin(ω_n t) with no decaying exponential envelope. In practice, real inductors and capacitors always have some parasitic resistance, so true ζ = 0 is impossible, but the analysis of the ideal lossless case explains the oscillatory behavior of high-Q circuits.
Question 4 True / False
A critically damped RLC circuit reaches its final value faster than an underdamped circuit with the same natural frequency ω_n.
TTrue
FFalse
Answer: False
A slightly underdamped circuit (e.g., ζ ≈ 0.7) reaches a threshold near the final value faster than a critically damped circuit because it overshoots. If the application only requires reaching within 10% of the final value, the underdamped circuit does so sooner — it arrives, overshoots past the threshold, and then rings back. Critical damping is the fastest response that approaches final value *monotonically* (without crossing it). The distinction matters enormously: a servo motor system targeting fast positioning can accept ζ ≈ 0.7 because slight overshoot is tolerable, and gains speed over critically damped control.
Question 5 Short Answer
Explain how the damping ratio ζ determines the character of an RLC transient response, and give one application where underdamped response is preferred and one where overdamped response is preferred.
Think about your answer, then reveal below.
Model answer: ζ = R/(2√(L/C)) is the ratio of resistive dissipation to reactive energy storage. ζ < 1: underdamped — energy sloshes between L and C, producing oscillation that decays. ζ = 1: critically damped — poles merge on negative real axis, fastest monotonic decay. ζ > 1: overdamped — poles are real and distinct, response is slow and monotonic.
Underdamped preferred: radio receiver tank circuits (ζ ≪ 1) need to resonate strongly at a target frequency to provide frequency selectivity; the ringing IS the function. Overdamped preferred: power supply output filters must suppress transients monotonically — any oscillation would appear as noise on the supply rail, corrupting downstream circuits. The engineering insight is that ζ is a design parameter chosen to match application requirements, not a quality metric where one extreme is universally better.