Fundamental Physics/Electricity/Electric oscillation wave circuit

 Electric current oscillation wave ${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int vdt=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}+{\frac {1}{T}}=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}=-{\frac {1}{T}}}$ ${\displaystyle i(t)=e^{\pm j{\sqrt {\frac {1}{T}}}t}=e^{\pm j\omega t}=ASin\omega t}$ ${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ Electric current oscillation standing wave ${\displaystyle Z_{L}-Z_{C}=0}$ ${\displaystyle Z_{C}=-Z_{L}}$ ${\displaystyle {\frac {1}{\omega C}}=-\omega L}$ ${\displaystyle \omega =\pm j{\sqrt {\frac {1}{LC}}}}$ ${\displaystyle V_{C}=-V_{L}}$ ${\displaystyle V(\theta )=ASin(\omega _{o}t+2\pi )-ASin(\omega _{o}t-2\pi )}$ Electric current decay oscillation wave ${\displaystyle L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0}$ ${\displaystyle {\frac {d^{2}i}{dt}}=-2\alpha {\frac {di}{dt}}-\beta i}$ ${\displaystyle \beta ={\frac {1}{T}}={\frac {1}{LC}}}$ ${\displaystyle \alpha =\beta \gamma ={\frac {R}{2L}}}$ ${\displaystyle T=LC}$ ${\displaystyle \gamma =RC}$ Root of equation 1 real root . ${\displaystyle \alpha =\beta }$ ${\displaystyle i=Ae^{-\alpha t}=A(\alpha )}$ 2 real roots . ${\displaystyle \alpha >\beta }$ ${\displaystyle i=Ae^{-\alpha \pm {\sqrt {\alpha -\beta }}t}}$ 2 complex roots . ${\displaystyle \alpha <\beta }$ ${\displaystyle i=Ae^{-\alpha \pm j{\sqrt {\beta -\alpha }}t}}$ ${\displaystyle i=Ae^{-\alpha t}e^{\pm j{\sqrt {\beta -\alpha }}t}}$ ${\displaystyle i=A(\alpha )Sin\omega t}$ ${\displaystyle A(\alpha )=Ae^{-\alpha t}}$ ${\displaystyle \omega ={\sqrt {\beta -\alpha }}}$ Peak electric current oscillation wave ${\displaystyle Z_{L}=-Z_{C}}$ ${\displaystyle \omega _{o}={\sqrt {\frac {1}{T}}}}$ ${\displaystyle T=LC}$ ${\displaystyle Z_{t}=R}$ ${\displaystyle i={\frac {v}{R}}}$ ${\displaystyle i(\omega =0)=0}$ ${\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}$ ${\displaystyle i(\omega =00)=0}$