 Underdamped RLC response

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 isFor initial conditions $q(0)=q_0$, $i(0)=0$,In the dimensionless example ${\omega }_0=1$, $\alpha =0.5$, the ringing angular frequency is ${\omega }_d=\sqrt{3}/2\approx 0.866025$, below the driven electric current resonance at ${\omega }_0=1$. The electric charge first crosses zero at $t=4\pi /(3\sqrt{3})\approx 2.41840$ and reaches its first negative extremum at $t=2\pi /\sqrt{3}\approx 3.62760$, where $q/q_0=-e^{-\pi /\sqrt{3}}\approx -0.163034$.

The sinusoidal factor changes sign while its envelope decays as $e^{-\alpha t}$. The total energy follows the exact balance $E^{\prime }=-R{i}^2$, rather than a pure exponential decay at every instant; in particular, its rate of loss is zero whenever $i=0$. As $\alpha \to 0$, the displayed solution tends to the lossless harmonic oscillator $q=q_0\cos ({\omega }_{0}t)$. As $\alpha \to {\omega }_0$ from below, $\sin ({\omega }_{d}t)/{\omega }_d\to t$, recovering the critically damped RLC response continuously. The overdamped RLC response reaches the same limit through $\sinh (\beta t)/\beta \to t$.