Electronics Fundamentals/Electronic Oscillator/Exponential Decreasing Amplitude Sinusoidal Wave Oscillator

Exponential Decreasing Amplitude Sinusoidal Wave Oscillator

Exponential Decreasing Amplitude Sinusoidal Wave Oscillator is an Electronics Device that has the capability to generate oscillation of an Exponential Decreasing Amplitude Sinusoidal Wave

Configuration

RLC connected in series

Mathematical Analysis

$L{\frac {di}{dt}}i+{\frac {1}{C}}\int idt+{\frac {1}{LC}}=0$

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

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

$s=(-\alpha \pm \lambda )t$

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

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

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

• $\lambda <0$

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

Summary

RLC series has the capability to generate oscillation of Exponential decreased sinusoidal wave when solving the characteristic equation to give complex roots i.e.

$\lambda <0$

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

$({\frac {R}{2L}})^{2}<({\frac {1}{LC}})^{2}$

$R<{\sqrt {\frac {L}{C}}}$