# Fundamental Physics/Motion/Electric Oscillations

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