Circuit Theory/LC Tuned Circuits

Series LC

A circuit of one Capacitor and one inductor connected in series

Circuit Impedance

$Z=Z_L+Z_C$

$Z=j\omega L+\frac{1}{j\omega C}$

$Z=\frac{1}{j\omega C}(j{\omega }^2+1)$

$Z=LC$

Natural Response

At equilibrium , the total voltage of the two components are equal to zero

$L\frac{dI}{dt}+IR=0$

$\frac{dI}{dt}=-I\frac{R}{L}$

$\int \frac{dI}{I}=-\frac{R}{L}\int dt$

$lnI=-\frac{t}{T}+C$

$I=e^{(}-\frac{t}{T}+C)$

$I=A{e}^{(}-\frac{t}{T})$

The Natural Response of the circuit is a Exponential Decrease in time

Resonance Response

$Z_L-Z_C=0$ . $V_L+V_C=0$

$\omega L=\frac{1}{\omega C}$

$\omega =\sqrt{\frac{1}{LC}}$

$V_C=-V_L$

In Resonance, Impedance of Inductor and Capacitance is equal and the sum of the Capacitor and Inductor's voltage are equal result in Standing Wave Oscillation . Therefore, Lossless LC series can generate Standing Wave Oscillation

LC in Parallel

A circuit of one Capacitor and one inductor connected in parallel

Circuit Impedance

$\frac{1}{Z}=\frac{1}{Z_L}+\frac{1}{Z_C}$

$Y=\frac{1}{j\omega L}+j\omega C$

$Y=\frac{1}{j\omega L}(j{\omega }^2+1)$