= Displacement and current bandwidths of a series RLC circuit
Write $\omega_0^2=1/(LC)$ and $\gamma=R/L$ for a <series RLC circuit>. The squared charge response has denominator $D=(\omega_0^2-\omega^2)^2+\gamma^2\omega^2$. Its nonzero <resonance> peak occurs at $\omega_r^2=\omega_0^2-\gamma^2/2$. The two squared half-height frequencies, when both exist, are
$$
\omega_\pm^2=\omega_0^2-\frac{\gamma^2}{2}\pm\gamma\sqrt{\omega_0^2-\frac{\gamma^2}{4}}.
$$
Their difference in <angular frequencies> is only approximately $\gamma$ at large <quality factor>. The <electric current> instead has $|I|^2=|V|^2/[R^2+(\omega L-1/(\omega C))^2]$. Its exact half-power frequencies satisfy $|\omega L-1/(\omega C)|=R$, whose positive roots differ by $R/L$, independently of <capacitance>. Thus an exact capacitance-independent <bandwidth> applies to current or resistor-voltage output; capacitor-voltage output is a low-pass response with a possible resonant peak.
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