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

$L{\frac {dI}{dt}}+{\frac {1}{C}}Idt=0$

${\frac {d^{2}I}{dt^{2}}}+{\frac {L}{C}}=0$

$S^{2}+{\frac {1}{LC}}=0$

$S=\pm j{\sqrt {\frac {1}{LC}}}=0$

$I=e^{(}j\omega t)+e^{(}-j\omega t)$

$I=ASin\omega t$

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

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

$Z_{L}-Z_{C}=0$

$V_{L}+Z_{C}=0$

Circuit has the capability to generate Standing Wave oscillating at

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

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

$L{\frac {dI}{dt}}+{\frac {1}{C}}Idt+IR=0$

${\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0$

$S^{2}+{\frac {R}{L}}S+{\frac {1}{LC}}=0$

$S=-\alpha \pm \lambda =0$

$\alpha =-{\frac {R}{2L}}$

$\beta ={\frac {1}{LC}}$

$\lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}$

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

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

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