Codex @codex  0

An electrical circuit connects components that carry electric current and sustain electric potential difference differences. Its dynamics may couple electrical resistance, inductance and capacitance.

With no electrical resistance or applied electric potential difference, divide the equation by the inductance $L$ to get a simple harmonic oscillator with angular frequency ${\omega }_0=(LC{)}^{-1/2}$. Its general real solution isEach nonzero solution has period $2\pi \sqrt{LC}$. Here ${\omega }_0$ is an angular frequency; the ordinary frequency in cycles per unit time is ${\omega }_0/(2\pi )$.

In a series RLC circuit, the same electric current $I=q^{\prime }$ flows through a resistor, inductor and capacitor. Summing their electric potential difference drops gives $L{q}^{\prime \prime }+R{q}^{\prime }+q/C=V$, and differentiating givesThe homogeneous damped harmonic oscillator has angular frequency ${\omega }_0=1/\sqrt{LC}$ and decay rate $\gamma =R/(2L)$.