Electronics Handbook/Devices/Oscillator/Passive Oscillator

Sinusoidal Wave Oscillator
Consider a series circuit of L and C connected in series
 * $$L \frac{dI}{dt} + \frac{1}{C} I dt = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{L}{C} = 0$$
 * $$S^2 + \frac{1}{LC} = 0$$
 * $$S = \pm j \sqrt{\frac{1}{LC}} t = \pm \omega t$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$
 * $$I = e^(St) = e^(j\omega t) + e^(-j\omega t)$$
 * $$I = A Sin \omega t$$
 * FreqWave1.png

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 capability to generate Standing Wave oscillating at


 * Standing_Wave.PNG

Exponential Decay Sinusoidal Wave Oscillator
Consider circuit of RLC connected in series
 * $$L \frac{dI}{dt} + \frac{1}{C} I dt + IR = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$S^2 + \frac{R}{L} S + \frac{1}{LC} = 0$$
 * $$S = (-\alpha \pm \lambda) t$$
 * $$i = e^(-\alpha \pm \lambda) t$$


 * $$\alpha = -\frac{R}{2L}$$
 * $$\beta = \frac{1}{LC}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$

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

 * 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