Crystal oscillator

Crystal oscillator


Crystal oscillator is used for high frequency application .It is basically a tuned circuit using a pizzo electric crystals as the resonant tank circuits.
A piezzo electrical crystal displays piezo electronic property.It is the ability to transform mechanical deformation into electricals charge and vice versa .Whwn the crystal is squazed it develope voltage and if a voltage is applied across  it a change in materials dimension results.
The most common piezzo electric materials are Rochelle salt and quartz
the resonant frequency and Q of a crystal depend upon the crystal dimension ,ie how the surface are oriented with respect to it axis and how the devices are m=mounted thinner and more frogile crystal are needed for high frequency oscillations
 
If the  resistance R is regulated the impedance of the crystal is purely reluctance

$iX=\frac{(jωL+\frac{1}{jωc} ) ( \frac{1}{jωc} )}{jωL+\frac{1}{jωc}+\frac{1}{jωc^`}}$

$=\frac{1-ω^2LC}{jωc^`-jω^3Lcc^`+jωc)}$
$=\frac{1-ω^2LC}{jω(c^`+c)-jω^3Lcc^`}$

let the reactance be zero at $ω=ω_s$ where $ω_s$ is series resonant frequency

$1-ω^2LC=0$ =>

$ω^2_s=\frac{1}{LC}$
$ω_s=\sqrt{\frac{1}{LC}}$
$f_s=\frac{1}{2π\sqrt{LC}}$
let the reactance become ∞ at $ω=ω_p$ where
$ω_p=$paralel resonant frequency
at $ω=ω_p$

 $=jω_p(c^`+c)=jω^3Lcc^`$
$c^`+c=jω^2_pLcc^`$
$ω^2_p=\frac{Lcc^`}{c^`+c}$
$ω_p=\sqrt{\frac{Lcc^`}{c^`+c}}$   
$f_p=\frac{1}{2π\sqrt{\frac{Lcc^`}{c^`+c}}}$  
 The circuits can be oscillate at a frequency between $ω_s$ and $ω_p$ .The oscillate frequency is determined by the crystal and not by rest of the circuit