Electronics Handbook/Components/Oscillator

Sin Wave Oscillator
The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series
 * $$L \frac{dI}{dt} + \frac{1}{C} I dt = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{L}{C} = 0$$
 * $$S^2 + \frac{1}{LC} = 0$$
 * $$S = \pm j \sqrt{\frac{1}{LC}} = 0$$
 * $$I = e^(j\omega t) + e^(-j\omega t)$$
 * $$I = A Sin \omega t$$
 * $$\omega = \sqrt{\frac{1}{LC}}$$

Standing Wave Oscillator
The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series operating at resonance
 * $$Z_L - Z_C = 0$$
 * $$V_L + Z_C = 0$$

Circuit has the capability to generate Standing Wave oscillating at
 * $$\omega = \sqrt{\frac{1}{LC}}$$

Sin Wave with Decresing Amplitude Oscillator
The circuit configuration of a Sin Wave Oscillation is made from connecting two components L and C in series
 * $$L \frac{dI}{dt} + \frac{1}{C} I dt + IR = 0$$
 * $$\frac{d^2I}{dt^2} + \frac{R}{L} \frac{dI}{dt} + \frac{1}{LC} = 0$$
 * $$S^2 + \frac{R}{L} S + \frac{1}{LC} = 0$$
 * $$S = -\alpha \pm \lambda = 0$$
 * $$\alpha = -\frac{R}{2L}$$
 * $$\beta = \frac{1}{LC}$$
 * $$\lambda = \sqrt{\alpha^2 - \beta^2}$$

When $$\alpha^2 < \beta^2$$
 * $$I = e^(-\alpha t) [e^(j\omega t) + e^(-j\omega t)]$$

The circuit has the ability to generate Sin Wave with decreasing amplitude