# Circuit Theory/LC Tuned Circuits

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## 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

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

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