# 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 capabilty to generate oscillation of an Exponential Decreasing Amplitude Sinusoidal Wave

### Configuration

RLC connected in series

## Mathematical Analysis

${\displaystyle L{\frac {di}{dt}}i+{\frac {1}{C}}\int idt+{\frac {1}{LC}}=0}$
${\displaystyle {\frac {d^{2}i}{dt^{2}}}++{\frac {R}{L}}{\frac {di}{dt}}+{\frac {1}{LC}}=0}$
${\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0}$
${\displaystyle s=(-\alpha \pm \lambda )t}$
${\displaystyle \alpha ={\frac {R}{2L}}}$
${\displaystyle \beta ={\frac {1}{LC}}}$
${\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}$

• ${\displaystyle \lambda <0}$
${\displaystyle i=e^{(}-\alpha t)[e^{(}j\omega t)+e^{(}-j\omega t)]}$

## 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.

${\displaystyle \lambda <0}$
${\displaystyle \alpha ^{2}<\beta ^{2}}$
${\displaystyle ({\frac {R}{2L}})^{2}<({\frac {1}{LC}})^{2}}$
${\displaystyle R<{\sqrt {\frac {L}{C}}}}$