Practical Electronics/Decreasing Sin Wave Oscillator

With an RLC series
 * $$L \frac{dI}{dt} + \frac{1}{C} \int I dt + IR = 0$$
 * $$\frac{d^2I}{dt} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$s = -(\alpha \pm \sqrt{\alpha^2 - \beta^2}) t = -(\alpha \pm \lambda) t$$
 * $$I = e^ -\alpha t (e^\lambda t + e^-\lambda t) $$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2} $$


 * $$\lambda = \sqrt{\alpha^2 - \beta^2} = 0 . $$
 * $$I = e^ -\alpha t $$


 * $$\lambda = \sqrt{\alpha^2 - \beta^2} > 0 . $$
 * $$I = e^ -\alpha t (e^\lambda t + e^-\lambda t) $$
 * $$I = A Cos \lambda t $$
 * $$A = \frac{e^ -\alpha t}{2} $$


 * $$\lambda = \sqrt{\alpha^2 - \beta^2} < 0 . $$
 * $$I = A(e^j\lambda t + e^-j\lambda t) $$
 * $$I = A Sin \lambda t $$
 * $$A = \frac{e^ -\alpha t}{2j} $$

The response of an LC series is a sinusoidal wave, or LC series can be used to produce decreasing sine wave oscillator.