Practical Electronics/Decreasing Sin Wave Oscillator

With an RLC series

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

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

$s=-(\alpha \pm \sqrt{{\alpha }^2-{\beta }^2})t=-(\alpha \pm \lambda )t$

$I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)$

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

• $\lambda =\sqrt{{\alpha }^2-{\beta }^2}=0.$

$I=e^{-}\alpha t$

• $\lambda =\sqrt{{\alpha }^2-{\beta }^2}>0.$

$I=e^{-}\alpha t(e^{\lambda }t+e^{-}\lambda t)$

$I=ACos\lambda t$

$A=\frac{e^{-}\alpha t}{2}$

• $\lambda =\sqrt{{\alpha }^2-{\beta }^2}<0.$

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

$I=ASin\lambda t$

$A=\frac{e^{-}\alpha t}{2j}$

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