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

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


 * $$\lambda < 0$$
 * $$i = e^(-\alpha t) [ e^(j\omega t) + e^(-j\omega t)]$$
 * [[File:SquareWaveFunctionFrequencyIncreasingToTheRightComparedWithTheSameFunctionSmoothedByAGaussian-de.svg|200px]]

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