# Fundamental Physics/Electronics/LC Circuit

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## Series LC ## Equilibrium Response

Circuit's Natural Response at equilibrium

$V_{L}+V_{C}=0$ $L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt=0$ ${\frac {d^{2}}{dt^{2}}}i(t)+{\frac {1}{LC}}i(t)=0$ $i(t)=Ae^{\pm j\omega t}=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$ The Natural Response at equilibrium of the circuit is a Sinusoidal oscillation wave The characteristic of the circuit can be expressed mathematically as Oscillation equation

${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}i(t)$ $T=LC$ Wave function

$i(t)=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ ## Resonance Response

At Resonance, The total Circuit's impedance is zero and the total volages are zero

$Z_{L}-Z_{C}=0$ $j\omega L={\frac {1}{j\omega C}}$ $\omega =\pm j{\sqrt {\frac {1}{T}}}$ $T=LC$ $V_{L}+V_{C}=0$ $V_{L}=-V_{C}$ $V(\theta )=ASin(\omega t+2\pi )-ASin(\omega t-2\pi )$ The Resonance Reponse of the circuit at resonance is a Standing (Sinusoidal) Wave