Theory: Damped Oscillations in a Series RLC Circuit
1) Governing equation
For an undriven series RLC circuit (no external voltage source), Kirchhoff’s loop rule gives:
\[
V_L+V_R+V_C=0
\quad\Rightarrow\quad
L\frac{dI}{dt}+RI+\frac{Q}{C}=0
\]
Using \(I=\dfrac{dQ}{dt}\), this becomes the standard second-order ODE:
\[
L\ddot Q + R\dot Q + \frac{1}{C}Q=0
\]
2) Key parameters
\[
\omega_0=\frac{1}{\sqrt{LC}},\qquad
\alpha=\frac{R}{2L},\qquad
\zeta=\frac{\alpha}{\omega_0}=\frac{R}{2\sqrt{L/C}}
\]
Here \(\omega_0\) is the natural (undamped) angular frequency, \(\alpha\) is the damping rate, and \(\zeta\) is the damping ratio.
3) Regimes
- Underdamped (\(\alpha<\omega_0\), or \(\zeta<1\)): oscillatory decay
- Critical damping (\(\alpha=\omega_0\), or \(\zeta=1\)): fastest return without oscillation
- Overdamped (\(\alpha>\omega_0\), or \(\zeta>1\)): non-oscillatory, slower return
4) Solutions (free response)
Underdamped
\[
\omega_d=\sqrt{\omega_0^2-\alpha^2}
\]
\[
Q(t)=e^{-\alpha t}\left(A\cos(\omega_d t)+B\sin(\omega_d t)\right)
\]
With initial conditions \(Q(0)=Q_0\) and \(I(0)=I_0\), one convenient choice is
\(A=Q_0\) and \(B=\dfrac{I_0+\alpha Q_0}{\omega_d}\).
The current is \(I(t)=\dot Q(t)\).
Critical damping
\[
Q(t)=(A+Bt)e^{-\alpha t}
\]
For \(Q(0)=Q_0\), \(I(0)=I_0\): \(A=Q_0\) and \(B=I_0+\alpha Q_0\).
Overdamped
\[
\beta=\sqrt{\alpha^2-\omega_0^2},\qquad s_{1,2}=-\alpha\pm\beta
\]
\[
Q(t)=c_1 e^{s_1 t}+c_2 e^{s_2 t}
\]
Constants \(c_1,c_2\) are determined from \(Q(0)=Q_0\) and \(I(0)=I_0\).
5) Voltages and a built-in consistency check
The element voltages are
\[
V_C=\frac{Q}{C},\qquad V_R=RI,\qquad V_L=L\frac{dI}{dt}
\]
For an ideal undriven series RLC circuit, the loop sum must be zero:
\[
V_L+V_R+V_C=0
\]
The calculator includes a “KVL residual” plot option to confirm this numerically.
6) Sample (from the prompt)
With \(R=10\ \Omega\), \(L=1\ \mathrm{H}\), \(C=1\,\mu\mathrm{F}=10^{-6}\ \mathrm{F}\):
\[
\omega_0=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{10^{-6}}}\approx 1000\ \mathrm{rad/s},\qquad
\alpha=\frac{R}{2L}=5\ \mathrm{s^{-1}}
\]
Since \(\alpha \ll \omega_0\), the response is underdamped and the plot shows a decaying sinusoid.
7) University extension (driven resonance)
With a sinusoidal driving voltage \(V(t)=V_0\cos(\omega t)\), the ODE becomes
\[
L\ddot Q + R\dot Q + \frac{1}{C}Q = V_0\cos(\omega t)
\]
Driven response introduces resonance, phase shift, and steady-state amplitude curves vs. \(\omega\).
