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

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

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

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

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

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

$I=ASin\omega t$

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

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

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

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

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

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

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

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

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

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