# Fundamental Physics/Electricity/Electric oscillation wave circuit

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