= Underdamped RLC response
{title2=$\omega_d=\sqrt{\omega_0^2-\alpha^2}$}
An <underdamped RLC response> is a free <series RLC circuit> response with $0<R<2\sqrt{L/C}$, equivalently $0<\alpha<\omega_0$. Its <characteristic roots> are $-\alpha\pm\mathrm{i}\omega_d$, and its general <electric charge> is
$$
q(t)=e^{-\alpha t}\left(A\cos(\omega_dt)+B\sin(\omega_dt)\right).
$$
For <initial conditions> $q(0)=q_0$, $i(0)=0$,
$$
q(t)=q_0e^{-\alpha t}\left(\cos(\omega_dt)+\frac\alpha{\omega_d}\sin(\omega_dt)\right),\qquad
i(t)=-\frac{q_0\omega_0^2}{\omega_d}e^{-\alpha t}\sin(\omega_dt).
$$
In the dimensionless example $\omega_0=1$, $\alpha=0.5$, the ringing <angular frequency> is $\omega_d=\sqrt3/2\approx0.866025$, below the driven <electric current> <resonance> at $\omega_0=1$. The <electric charge> first crosses zero at $t=4\pi/(3\sqrt3)\approx2.41840$ and reaches its first negative extremum at $t=2\pi/\sqrt3\approx3.62760$, where $q/q_0=-e^{-\pi/\sqrt3}\approx-0.163034$.
The sinusoidal factor changes sign while its envelope decays as $e^{-\alpha t}$. The total <energy> follows the exact balance $E'=-Ri^2$, rather than a pure <exponential decay> at every instant; in particular, its rate of loss is zero whenever $i=0$. As $\alpha\to0$, the displayed solution tends to the lossless <harmonic oscillator> $q=q_0\cos(\omega_0t)$. As $\alpha\to\omega_0$ from below, $\sin(\omega_dt)/\omega_d\to t$, recovering the <critically damped RLC response> continuously. The <overdamped RLC response> reaches the same limit through $\sinh(\beta t)/\beta\to t$.
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