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.