A series RLC circuit has a very large resistor R. After a switch closes, the capacitor voltage response will be:
AAn underdamped decaying sinusoid, because the circuit contains both L and C
BA sum of two decaying exponentials (overdamped), because large R makes α > ω₀
CA critically damped response, which is the default for series RLC circuits
DA sustained sinusoidal oscillation, because LC circuits naturally oscillate
The response type depends on the damping ratio ζ = α/ω₀, where α = R/(2L) for series RLC. A very large R makes α large, so α > ω₀ = 1/√(LC), giving ζ > 1: overdamped. The response is a sum of two decaying exponentials with no oscillation. Option D is the most common misconception: LC circuits can sustain oscillation only if R = 0. Any resistance dissipates energy and drives the response to a new steady state — oscillation decays, it does not persist.
Question 2 Multiple Choice
After finding the characteristic roots of a second-order circuit, what information is required to determine the unknown constants in the general solution?
AThe values of R, L, and C alone fully determine the constants
BThe forced (steady-state) response and the natural frequency ω₀
CTwo initial conditions: the initial value of the variable and the initial value of its time derivative
DOne initial condition: the initial energy stored in the capacitor or inductor
A second-order ODE has two free constants in its general solution (A₁ and A₂ for overdamped; B₁ and B₂ for underdamped). These require exactly two independent initial conditions. The first is v_C(0⁺), which cannot jump due to capacitor continuity. The second is dv_C/dt(0⁺), obtained by applying KVL or KCL at t = 0⁺ and using i_C = C dv_C/dt. Using only one initial condition — the most common error — leaves the system underdetermined.
Question 3 True / False
An underdamped RLC circuit, once disturbed, will oscillate at its natural frequency indefinitely if no further input is applied.
TTrue
FFalse
Answer: False
Underdamped (ζ < 1) means the response is a *decaying* sinusoid: e^(−αt)(B₁cos(ω_d t) + B₂sin(ω_d t)). The amplitude shrinks exponentially toward the forced (steady-state) response because the resistor continuously dissipates energy. Sustained oscillation would require zero resistance. The key distinction: underdamped means oscillation *occurs*, not that it persists forever.
Question 4 True / False
Finding the complete solution of a second-order transient circuit requires two initial conditions: the initial value of the circuit variable and the initial value of its time derivative.
TTrue
FFalse
Answer: True
A second-order ODE has two free constants in its complementary solution, requiring exactly two independent conditions. The initial value of v_C or i_L is often read directly from the circuit at t = 0⁻ (continuity of stored energy). The initial derivative must be calculated using KVL/KCL at t = 0⁺. This two-condition requirement distinguishes second-order from first-order transient analysis, where only one initial condition suffices.
Question 5 Short Answer
Why does a critically damped circuit return to steady state faster than an overdamped circuit, even though neither oscillates?
Think about your answer, then reveal below.
Model answer: In an overdamped circuit (ζ > 1), there are two distinct real exponential modes, each decaying at a different rate. One mode always decays more slowly than the critically damped rate, so the response lingers. At critical damping (ζ = 1), the solution (A₁ + A₂t)e^(−αt) achieves the fastest possible monotonic approach to equilibrium. Increasing R beyond critical damping makes the two characteristic roots more unequal; the slower root becomes less negative and takes longer to decay, extending the transient.
This is the engineering motivation for critical damping in applications like servo controllers and suspension systems: it minimizes settling time without the overshoot that comes with underdamped response. Overdamping is 'too much damping' — excess resistance slows one exponential mode, lengthening the transient even though no oscillation occurs.