Electronics/Electronics Formulas/Series Circuits/Series LC

The total Impedance of the circuit

${\displaystyle Z=Z_{R}+Z_{L}}$

${\displaystyle Z=R+j\omega L}$

${\displaystyle Z={\frac {1}{R}}(1+j\omega T)}$

${\displaystyle T={\frac {L}{R}}}$

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

${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt=0}$

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

${\displaystyle s^{2}+{\frac {1}{T}}i=0}$

${\displaystyle s={\sqrt {-{\frac {1}{LC}}}}=\pm j{\sqrt {\frac {1}{LC}}}t=\pm j{\sqrt {\frac {1}{T}}}t=\pm j\omega t}$

${\displaystyle i=Ae^{st}=Ae^{\pm j\omega t}=Asin\omega t}$

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

${\displaystyle T=LC}$

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

${\displaystyle \omega _{o}L=-{\frac {1}{\omega _{o}C}}}$

${\displaystyle \omega _{o}={\sqrt {\frac {-1}{LC}}}}$

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

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

${\displaystyle V(\theta )=Vsin(\omega _{o}+\theta )-Vsin(\omega _{o}-\theta )}$

Series LC can be characterised by

2nd order Differential Equation

${\displaystyle {\frac {d^{2}i}{dt^{2}}}+{\frac {1}{T}}=0}$

${\displaystyle T=LC}$

With Natural Response of a Wave function

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

With Resonance Response of a Standing Wave function

${\displaystyle v=ASin(\omega _{o}t+\theta )-ASin(\omega _{o}t-\theta )}$