# Fundamental Physics/Electricity/Electric Oscillation

## Electric Oscillation

Electric wave oscillation has been observed in series RLC circuit

## RLC Series

Consider a circuit of 3 components L, C and R connected in series ### At equilibrium

$V_{L}+V_{C}+V_{R}=0$ $L{\frac {di}{dt}}+{\frac {1}{C}}\int idt+iR=0$ ${\frac {d^{2}i}{dt^{2}}}+{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}i=0$ ${\frac {d^{2}i}{dt^{2}}}+2\alpha {\frac {di}{dt}}+{\frac {\beta }{i}}=0$ Using laplace transform

$s^{2}+2\alpha s+\beta =0$ 1 real root . $\alpha =\beta$ $s=-\alpha$ $i=Ae^{-\alpha t}$ 2 real roots . $\alpha >\beta$ $s=-\alpha \pm {\sqrt {\alpha -\beta }}$ $i=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}$ 2 complex roots . $\alpha <\beta$ $s=-\alpha \pm j{\sqrt {\beta -\alpha }}$ $i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t$ With

$A(\alpha )=Ae^{-\alpha t}$ $\omega ={\sqrt {\beta -\alpha }}$ $\alpha ={\frac {R}{2L}}=\beta \gamma$ $\beta ={\frac {1}{LC}}={\frac {1}{T}}$ $\gamma =RC$ $T=LC$ ### At resonance

$Z_{L}+Z_{C}+Z_{R}=R$ $i={\frac {v}{R}}$ $Z_{L}+Z_{C}=0$ $Z_{L}=-Z_{C}$ $j\omega L={\frac {1}{j\omega C}}$ $\omega _{o}=\pm {\sqrt {\frac {1}{T}}}$ $i(\omega =0)=0$ . capacitor opens circuit
$i(\omega =\omega _{o})={\frac {v}{R}}$ . $Z_{L}=Z_{C}$ $i(\omega =00)=0$ . inductor opens circuit


## LC Series

Consider a circuit of 2 components L and C connected in series ### At equilibrium

$V_{L}+V_{C}=0$ $L{\frac {di}{dt}}+{\frac {1}{C}}\int idt=0$ ${\frac {d^{2}i}{dt^{2}}}+{\frac {1}{T}}i=0$ ${\frac {d^{2}i}{dt^{2}}}=-{\frac {1}{T}}i$ Using laplace transform

$s^{2}=-{\frac {1}{T}}$ $s=\pm j{\sqrt {\frac {1}{T}}}=\pm j\omega$ Root of equation above

$i=Ae^{st}=Ae^{\pm j\omega t}=ASin\omega t$ With

$\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$ ### At resonance

$Z_{L}+Z_{C}=0$ $Z_{L}=-Z_{C}$ $j\omega L={\frac {1}{j\omega C}}$ $\omega _{o}=\pm {\sqrt {\frac {1}{T}}}$ $V_{L}=-Z_{C}$ $V(\theta )=ASin(\omega _{o}t+2\pi )-ASin(\omega _{o}t-2\pi )$ ## Summary

 Configuration Operation Mode Osicallation Equation Wave Function Angular frequency Time Constant Wave Form LC series At equilibrium ${\frac {d^{2}i}{dt^{2}}}=-{\frac {1}{T}}$ $i=ASin\omega t$ $\omega ={\sqrt {\frac {1}{T}}}$ $T=LC$ Oscillation is of the Sinusoidal wave LC series At resonance $Z_{L}=-Z_{C}$ $V_{L}=-V_{C}$ $v(\theta )=ASin(\omega _{o}t+2\pi )-ASin(\omega _{o}t-2\pi )$ $\omega _{o}=\pm j{\sqrt {\frac {1}{T}}}$ $T=LC$ Oscillation is of the Sinusoidal standing wave RLC series At equilibrium ${\frac {d^{2}i}{dt^{2}}}=-2\alpha {\frac {d^{2}i}{dt^{2}}}-\beta i$ $i=A(\alpha )Sin\omega t$ $\omega ={\sqrt {\beta -\alpha }}$ $\beta ={\frac {1}{T}}={\frac {1}{LC}}$ $\alpha =\beta \gamma ={\frac {2R}{L}}$ $T=LC$ $\gamma =RC$ $T=LC$ Oscillation is of the Decay current sinusoidal oscillation wave RLC series At resonance $Z_{L}=-Z_{C}$ $Z_{t}=R$ $I(\omega =0)=0$ $I(\omega =\omega _{o})={\frac {v}{R}}$ $I(\omega =0)=0$ $\omega _{o}=\pm j{\sqrt {\frac {1}{T}}}$ $T=LC$ Peak current sinusoidal oscillation wave