# Electronics Handbook/Components/Oscillator

## Sin Wave Oscillator

The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series

${\displaystyle L{\frac {dI}{dt}}+{\frac {1}{C}}Idt=0}$
${\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {L}{C}}=0}$
${\displaystyle S^{2}+{\frac {1}{LC}}=0}$
${\displaystyle S=\pm j{\sqrt {\frac {1}{LC}}}=0}$
${\displaystyle I=e^{(}j\omega t)+e^{(}-j\omega t)}$
${\displaystyle I=ASin\omega t}$
${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

## Standing Wave Oscillator

The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series operating at resonance

${\displaystyle Z_{L}-Z_{C}=0}$
${\displaystyle V_{L}+Z_{C}=0}$

Circuit has the capability to generate Standing Wave oscillating at

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

## Sin Wave with Decresing Amplitude Oscillator

The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series

${\displaystyle L{\frac {dI}{dt}}+{\frac {1}{C}}Idt+IR=0}$
${\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0}$
${\displaystyle S^{2}+{\frac {R}{L}}S+{\frac {1}{LC}}=0}$
${\displaystyle S=-\alpha \pm \lambda =0}$
${\displaystyle \alpha =-{\frac {R}{2L}}}$
${\displaystyle \beta ={\frac {1}{LC}}}$
${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

When ${\displaystyle \alpha ^{2}<\beta ^{2}}$

${\displaystyle I=e^{(}-\alpha t)[e^{(}j\omega t)+e^{(}-j\omega t)]}$

The circuit has the ability to generate Sin Wave with decreasing amplitude