RLC Circuit Response Calculator

The implementation solves the constant-coefficient circuit ODEs exactly for t ≥ 0, with continuous capacitor voltage and inductor current. The equations below are derived from Kirchhoff's laws; Oregon State ENGR 202, Section 4, pp. 6–20 and 33–50, provides the series RLC solution and independent worked examples.

• RC: $C\dot v_C=(V_s-v_C)/R$, with $\tau=RC$.
• RL: $L\dot i_L=V_s-Ri_L$, with $v_R=Ri_L$ and $\tau=L/R$.
• Series RLC: $C\dot v_C=i_L$, $L\dot i_L=V_s-Ri_L-v_C$.
• Parallel RLC: $C\dot v=I_s-v/R-i_L$, $L\dot i_L=v$.

For RLC, $\omega_0=1/\sqrt{LC}$ and $\zeta=\alpha/\omega_0$. Series $\alpha=R/(2L)$; parallel $\alpha=1/(2RC)$. Both use $Q=1/(2\zeta)$. The homogeneous solution uses decaying sine/cosine terms, a repeated-pole solution at critical damping, or two real exponentials. The slow real pole is computed as $-\omega_0/(\zeta+\sqrt{\zeta^2-1})$ to avoid subtracting nearly equal numbers.

With $r=f/f_{char}$, RC/RL use $H=1/(1+jr)$. Series RLC uses $H=1/(1-r^2+j2\zeta r)$. Parallel RLC uses $H=Z/R=j2\zeta r/(1-r^2+j2\zeta r)$. Thus $f_0=\omega_0/(2\pi)$ is the series current resonance and parallel impedance resonance, while the series capacitor-voltage peak is $f_0\sqrt{1-2\zeta^2}$ only for $\zeta<1/\sqrt2$.

The curves use 1,601 time samples with quadratic spacing near switching and 1,201 logarithmic frequency samples. Playback evaluates only the cursor state and drawing; the curves do not depend on playback position.

Ideal, linear, time-invariant R, L and C; ideal constant source after an instantaneous step at t = 0; no external load. R is total series resistance for RC/RL/series RLC and a pure shunt resistance for parallel RLC. The arrows define positive voltage and current. Initial capacitor voltage and inductor current are independent stored states, not assumed to be the preceding DC equilibrium. The Kirchhoff equations and parallel step solution are derived here; the cited sources independently support the method and benchmark values. Bode response assumes zero initial energy. Electrical SI units have the same meaning in metric and US display modes.

Use positive finite components. RLC is restricted to $0.025\le\zeta\le20$ (equivalently $0.025\le Q\le20$) for readable sampled curves and numerical conditioning. Values within $10^{-8}$ of critical damping use the repeated-pole limit. Time and Bode plots are sampled; use the analytic cursor and peak result for exact point values. A decay time is not a guaranteed settling time, especially at critical damping or with arbitrary initial energy. Inactive component steps in “How it’s calculated” retain default values and apply only to their named topology. No winding resistance inside the parallel inductor, ESR, saturation, semiconductor switching, tolerances, thermal drift, or distributed parasitics are modeled. This tool does not size protection or certify component ratings.