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}}}

I=e^{-}\alpha t

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

I=ACos\lambda t

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

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.

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