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 volatge of the two components are equal to zero

$L\frac{dI}{dt} + IR = 0$
$\frac{dI}{dt} = - I \frac{R}{L}$
$\int \frac{dI}{I} = - \frac{R}{L} \int dt$
$ln I = - \frac{t}{T} + C$
$I = e^(- \frac{t}{T} + C)$
$I = A e^(- \frac{t}{T})$

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

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