Series LC

Equilibrium Response

Circuit's Natural Response at equilibrium

${\displaystyle V_{L}+V_{C}=0}$

${\displaystyle L{\frac {d}{dt}}i(t)+{\frac {1}{C}}\int i(t)dt=0}$

${\displaystyle {\frac {d^{2}}{dt^{2}}}i(t)+{\frac {1}{LC}}i(t)=0}$

${\displaystyle i(t)=Ae^{\pm j\omega t}=ASin\omega t}$

${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$

${\displaystyle T=LC}$

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

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

${\displaystyle T=LC}$

Wave function

${\displaystyle i(t)=ASin\omega t}$

${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$

Resonance Response

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

${\displaystyle Z_{L}-Z_{C}=0}$

${\displaystyle j\omega L={\frac {1}{j\omega C}}}$

${\displaystyle \omega =\pm j{\sqrt {\frac {1}{T}}}}$

${\displaystyle T=LC}$

${\displaystyle V_{L}+V_{C}=0}$

${\displaystyle V_{L}=-V_{C}}$

${\displaystyle V(\theta )=ASin(\omega t+2\pi )-ASin(\omega t-2\pi )}$

The Resonance Reponse of the circuit at resonance is a Standing (Sinusoidal) Wave