Electronics Handbook/Devices/Oscillator/Passive Oscillator

Consider a series circuit of L and C connected in series

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

${\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {L}{C}}=0}$

${\displaystyle S^{2}+{\frac {1}{LC}}=0}$

${\displaystyle S=\pm j{\sqrt {\frac {1}{LC}}}t=\pm \omega t}$

${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

${\displaystyle I=e^{(}St)=e^{(}j\omega t)+e^{(}-j\omega t)}$

${\displaystyle I=ASin\omega t}$

The circuit of series L and C operates in resonance when the impedance of the two components cancel out

${\displaystyle Z_{L}-Z_{C}=0}$

${\displaystyle \omega L={\frac {1}{\omega C}}}$

${\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}$

${\displaystyle V_{L}+Z_{C}=0}$

${\displaystyle Z_{C}=-V_{L}}$

Circuit has the capability to generate Standing Wave oscillating at

Consider circuit of RLC connected in series

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

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

${\displaystyle S^{2}+{\frac {R}{L}}S+{\frac {1}{LC}}=0}$

${\displaystyle S=(-\alpha \pm \lambda )t}$

${\displaystyle i=e^{(}-\alpha \pm \lambda )t}$

${\displaystyle \alpha =-{\frac {R}{2L}}}$

${\displaystyle \beta ={\frac {1}{LC}}}$

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

When ${\displaystyle \lambda <0}$

${\displaystyle \alpha ^{2}<\beta ^{2}}$

${\displaystyle i=e^{(}-\alpha \pm j\lambda )t}$

${\displaystyle i=e^{(}-\alpha t)[e^{(}j\lambda t)+e^{(}-j\lambda t)]}$

${\displaystyle i=e^{(}-\alpha t)Sin\lambda t}$

The circuit has the ability to generate Exponenential Decreasing Amplitude Sinusoidal Wave

1. Lossless LC series operates at Equililibrium has the capabilities to generate Sinusoidal Wave
2. Lossless LC series operates at Resonance has the capablities to generate Standing Sinusoidal Wave
3. Lossy RLC series operates at Equililibrium has the capablities to generate Exponential Decrese Sinusoidal Wave