# 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

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

Using laplace transform

${\displaystyle s^{2}+2\alpha s+\beta =0}$

1 real root . ${\displaystyle \alpha =\beta }$

${\displaystyle s=-\alpha }$
${\displaystyle i=Ae^{-\alpha t}}$

2 real roots . ${\displaystyle \alpha >\beta }$

${\displaystyle s=-\alpha \pm {\sqrt {\alpha -\beta }}}$
${\displaystyle i=Ae^{(-\alpha \pm {\sqrt {\alpha -\beta }})t}}$

2 complex roots . ${\displaystyle \alpha <\beta }$

${\displaystyle s=-\alpha \pm j{\sqrt {\beta -\alpha }}}$
${\displaystyle i=Ae^{(-\alpha \pm j{\sqrt {\beta -\alpha }})t}=A(\alpha )Sin\omega t}$

With

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

### At resonance

${\displaystyle Z_{L}+Z_{C}+Z_{R}=R}$
${\displaystyle i={\frac {v}{R}}}$
${\displaystyle Z_{L}+Z_{C}=0}$
${\displaystyle Z_{L}=-Z_{C}}$
${\displaystyle j\omega L={\frac {1}{j\omega C}}}$
${\displaystyle \omega _{o}=\pm {\sqrt {\frac {1}{T}}}}$


${\displaystyle i(\omega =0)=0}$ . capacitor opens circuit
${\displaystyle i(\omega =\omega _{o})={\frac {v}{R}}}$ . ${\displaystyle Z_{L}=Z_{C}}$
${\displaystyle i(\omega =00)=0}$ . inductor opens circuit


## LC Series

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

### At equilibrium

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

Using laplace transform

${\displaystyle s^{2}=-{\frac {1}{T}}}$
${\displaystyle s=\pm j{\sqrt {\frac {1}{T}}}=\pm j\omega }$

Root of equation above

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


With

${\displaystyle \omega ={\sqrt {\frac {1}{T}}}}$
${\displaystyle T=LC}$


### At resonance

${\displaystyle Z_{L}+Z_{C}=0}$
${\displaystyle Z_{L}=-Z_{C}}$
${\displaystyle j\omega L={\frac {1}{j\omega C}}}$
${\displaystyle \omega _{o}=\pm {\sqrt {\frac {1}{T}}}}$

${\displaystyle V_{L}=-Z_{C}}$
${\displaystyle V(\theta )=ASin(\omega _{o}t+2\pi )-ASin(\omega _{o}t-2\pi )}$


## Summary

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