# Circuit Theory/LC Tuned Circuits

## Series LC

A circuit of one Capacitor and one inductor connected in series

### Circuit Impedance

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

### Natural Response

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

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

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

### Resonance Response

${\displaystyle Z_{L}-Z_{C}=0}$  . ${\displaystyle V_{L}+V_{C}=0}$
${\displaystyle \omega L={\frac {1}{\omega C}}}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$
${\displaystyle V_{C}=-V_{L}}$

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

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