Electronics Handbook/Devices/Oscillator/Passive Oscillator

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Sinusoidal Wave Oscillator

Consider a series circuit of L and C connected in series

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

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

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

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

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

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

I=ASin\omega t

Standing Wave Oscillator

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

Z_{L}-Z_{C}=0

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

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

V_{L}+Z_{C}=0

Z_{C}=-V_{L}

Circuit has the capaability to generate Standing Wave oscillating at

Exponential Decay Sinusoidal Wave Oscillator

Consider circuit of RLC connected in series

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

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

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

S=(-\alpha \pm \lambda )t

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

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

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

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

When $\lambda <0$

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

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

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

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

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

Summary

Lossless LC series operates at Equililibrium has the capablities to generate Sinusoidal Wave
Lossless LC series operates at Resonance has the capablities to generate Standing Sinusoidal Wave
Lossy RLC series operates at Equililibrium has the capablities to generate Exponential Decrese Sinusoidal Wave

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