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RLC Second-Order Transient Response · Step Response and Damping Characteristics

Series RLC circuit step response under square wave excitation | Adjust \(R, L, C\) to observe underdamped (oscillatory decay), critically damped, and overdamped states

Circuit Parameter Adjustment

50 Ω
10 Ω500 Ω
10.0 mH
1 mH50 mH
10.0 μF
1 μF100 μF
50 Hz
10 Hz200 Hz

Characteristic Parameters

\(\zeta\) Damping Ratio --
Damping State --
\(\omega_0\) Resonant Frequency --
\(\omega_d\) Damped Frequency --

Series RLC Circuit · Square Wave Excitation Response

\(v_{\text{in}}(t)\), \(v_C(t)\), \(i(t)\)
Gray staircase: input square wave \(v_{\text{in}}\); Blue curve: capacitor voltage \(v_C(t)\); Orange curve: current \(i(t)\) (scaled for display)

Response Metrics

Peak Voltage
--
Overshoot
--
Settling Time (est.)
--

Three Damping States

Underdamped (\(\zeta < 1\)): \(v_C(t)\) oscillates with decay, overshoot increases as \(\zeta\) decreases. Poles are a pair of complex conjugates.
Critically Damped (\(\zeta = 1\)): Fastest non-overshooting response, \(v_C(t)\) rises monotonically without oscillation. Poles are repeated real roots.
Overdamped (\(\zeta > 1\)): Response rises slowly and monotonically, no oscillation. Poles are two distinct negative real roots.

Characteristic Formulas

\(\zeta = \frac{R}{2}\sqrt{\frac{C}{L}}\), \(\omega_0 = \frac{1}{\sqrt{LC}}\)
\(\omega_d = \omega_0\sqrt{1-\zeta^2}\) (underdamped)
Overshoot \(M_p = e^{-\pi\zeta/\sqrt{1-\zeta^2}} \times 100\%\)
Poles: \(s_{1,2} = -\zeta\omega_0 \pm \omega_0\sqrt{\zeta^2-1}\)