= Critically damped RLC response
{title2=$\alpha=\omega_0,\quad R=2\sqrt{L/C}$}
A <critically damped RLC response> is the boundary between <underdamped RLC response> and <overdamped RLC response>. The free <series RLC circuit> has a repeated <characteristic root> $-\omega_0$, with general solution
$$
q(t)=(A+Bt)e^{-\omega_0t}.
$$
For <initial conditions> $q(0)=q_0$, $i(0)=0$,
$$
q(t)=q_0(1+\omega_0t)e^{-\omega_0t},\qquad
i(t)=-q_0\omega_0^2t e^{-\omega_0t}.
$$
In the plotted normalization, $q(1)=2/e\approx0.735759$ and $i(1)=-1/e\approx-0.367879$. The <electric charge> returns without ringing; the <polynomial> factor $1+\omega_0t$ matters, so a single <exponential function> does not give the whole response.
For fixed $L,C$ and these <initial conditions>, <critical damping> avoids ringing while giving a faster asymptotic return than any larger $R$: the <overdamped RLC response> has a root closer to zero. This makes <critical damping> a useful tuning target for settling after switching. An MIT laboratory exercise makes the distinction observable by asking students to compare all three cases with $L=47\,\mathrm{mH}$ and $C=47\,\mathrm{nF}$. Those values give $R_{\mathrm{crit}}=2\sqrt{L/C}=2000\,\Omega$ and $\omega_0\approx21276.6\,\mathrm{s}^{-1}$. https://ocw.mit.edu/courses/6-071j-introduction-to-electronics-signals-and-measurement-spring-2006/491e7e50ee65521c601ee51d3c6f7e5f_lab15_transients.pdf[MIT 6.071, Spring 2006, Laboratory 15, exercise 2]{ref}.
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