Practical Electronics/Decreasing Sin Wave Oscillator

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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.