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

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

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

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

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

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

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

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

${\frac {1}{Z}}={\frac {1}{Z_{L}}}+{\frac {1}{Z_{C}}}$

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

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