Fundamental Physics/Motion/Electric Oscillations

 Series LC At Equilibrium At Resonance $V_{L}+V_{C}=0$ $Z_{L}+Z_{C}=0$ $j\omega L+{\frac {1}{j\omega C}}=0$ $\omega _{o}=\pm j{\sqrt {\frac {1}{LC}}}$ Oscillation Equation ${\frac {d^{2}}{dt^{2}}}i(t)=-{\frac {1}{T}}i(t)$ $V(\theta )=V_{L}+V_{C}$ Wave Function $i(t)=ASin\omega t$ $V(\theta )=ASin(\omega _{o}t+2\pi )$ $-ASin(\omega _{o}t-2\pi )$ Angular Velocity $\omega ={\sqrt {\frac {1}{T}}}$ $\omega _{o}={\sqrt {\frac {1}{T}}}$ Time Constant $T=LC$ $T=LC$ Series LC At Equilibrium At Resonance $V_{L}+V_{C}+V_{R}=0$ $Z_{L}+Z_{C}=0$ $Z_{t}=Z_{L}+Z_{C}+Z_{R}=R$ Oscillation Equation ${\frac {d^{2}}{dt^{2}}}i(t)=-2\alpha {\frac {d^{2}}{dt^{2}}}i(t)-\beta i(t)$ $i(\omega =\omega _{o})={\frac {v}{R}}$ Wave Function $i(t)=A(\alpha )Sin\omega t$ $i(\omega =0)=0.i(\omega =\omega _{o})={\frac {v}{R}}.i(\omega =00)=0$ Angular Velocity $\omega ={\sqrt {\beta -\alpha }}$ $\omega _{o}={\sqrt {\frac {1}{T}}}$ Time Constant $T=LC$ $T=LC$ β $\beta ={\frac {1}{T}}$ α $\alpha =\beta \gamma$ γ $\gamma =RC$ 