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