Second-order RLC circuits

Second order: two numbers decide everything

• A series RLC has natural frequency $ω_{0} = 1/\sqrt{LC}$ and damping $α = R/(2L)$; one comparison picks the regime — overdamped ($α > ω_{0}$, two real decays), critically damped ($α = ω_{0}$, fastest return without ringing), underdamped ($α < ω_{0}$, ringing at $ω_{d} = \sqrt{ω_{0}^{2} - α^{2}}$). The Q factor $ω_{0}/(2α)$ says how long it rings.
• Two energy stores give two constants, so two endpoint checks are free: initial value and initial slope (no instantaneous current through L, no instantaneous voltage jump on C).

{exercise:uni-ee4x-second}