= Spark-excited damped oscillation
{title2=$q(t)=Ae^{-\alpha t}\cos(\omega_dt+\phi)$}
A <spark gap> can rapidly discharge a charged <capacitor> through an <inductor>. After excitation, the stored <energy> runs down through <Joule heating>, radiation and other losses. A key held down can permit repeated discharges: each discharge starts a damped train of rapid <oscillations>. The spark repetition <frequency>, the electrical <oscillation> <frequency> within each train, and the rate of keyed symbols are different quantities. Repetition does not make each train a maintained <sinusoid>.
An original lumped model replaces the conducting <spark gap> and other losses by a constant total <electrical resistance> $R>0$ in an isolated <series RLC circuit>. For <electric charge> $q$ on the <capacitor>, the <Kirchhoff cycle law> gives
$$
L\ddot q+R\dot q+\frac qC=0,\qquad
\omega_0=\frac1{\sqrt{LC}},\quad
\alpha=\frac R{2L},\quad
Q=\frac{\omega_0L}R.
$$
For $Q>1/2$, its characteristic roots are $-\alpha\pm\mathrm i\omega_d$, with $\omega_d=\sqrt{\omega_0^2-\alpha^2}$, proving the stated damped form. The <electric current> $\dot q$ has the same exponential decay scale. Its <wave envelope> falls by $\exp(-2\pi\alpha/\omega_d)$ per cycle; the <energy> envelope falls twice as fast in the exponent. A maintained <sinusoid> instead requires continuing input <power> balancing losses.
With $L=5\,\mathrm{mH}$, $C=2\,\mathrm{nF}$ and $R=100\,\Omega$, the model gives $f_0=\omega_0/(2\pi)=50.329\,\mathrm{kHz}$, <Q factor> $Q=15.811$, amplitude <time constant> $1/\alpha=100\,\mathrm{\mu s}$, and damped <frequency> $\omega_d/(2\pi)=50.304\,\mathrm{kHz}$. These are chosen component values. A real <spark gap> switches nonlinearly, an <antenna> radiates, and coupling exchanges <energy> between circuits; one constant-$R$ free decay does not model the complete transmitter. Rapid <damping> also broadens the <frequency spectrum>, which helps explain why short trains make sharp receiver selection difficult.
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