Parallel RLC circuit ($Y(\omega )$)

An ideal resistor, inductor and capacitor connected across the same voltage have admittance $Y=1/R+i(\omega C-1/(\omega L))$ in the $e^{i\omega t}$ convention. At ${\omega }_0=1/\sqrt{LC}$ the reactive branch currents cancel, giving minimum source current $V_0/R$ and maximum impedance $R$. This differs from a lossy series coil placed across a capacitor.

With no external current, Kirchhoff's current law is $C\dot{V}+V/R+I_L=0$, and $L{\dot{I}}_L=V$. Differentiating gives the title equation. For $4R^2>L/C$, its real solutions are $A{e}^{-t/(2RC)}\cos ({\omega }_{d}t+\varphi )$ with ${\omega }_d^2=1/(LC)-1/(4R^2{C}^2)$. The original current balance fixes the compatible initial inductor current. A single complex exponential represents a physical voltage through its real part; general complex solutions require both conjugate characteristic roots.

When $R_s$ is in series with a parallel RLC circuit, the ratio of parallel-network voltage to voltage across $R_s$ is its impedance divided by $R_s$. Its peak is $R/R_s$, and its phase passes from $+\pi /2$ through zero to $-\pi /2$ as angular frequency rises through $1/\sqrt{LC}$. This is not the transfer ratio relative to the total input voltage, which includes the additional series voltage drop and has peak $R/(R+R_s)$. Identifying voltage endpoints is necessary before using the voltage divider formula.

For the ratio of resistor-current amplitude to total-current amplitude to exceed $1/\sqrt{2}$, $|\omega C-1/(\omega L)|<1/R$. The exact positive endpoint difference is $\Delta \omega =1/(RC)$ and the quality factor is $Q=R\sqrt{C/L}$. Greater shunt resistance means weaker dissipation, unlike increasing a series resistance.