Sep 182014
 

The Crystal Oscillators (Piezoelectric Oscillator)

A quartz crystal has a property called: piezoelectric effect. This effect causes, by applying a mechanical pressure on the surface of the crystal, that a voltage is developed on the opposite sides. In a similar way, a voltage applied on the faces of the crystal produces a mechanical distortion on their surface.

An AC voltage causes mechanical vibration whose natural frequency is very stable and depends on the nature and carved crystal.

Crystal Oscillator equivalent circuit

The equivalent circuit model of a quartz crystal (look at the picture below) is characterized by a very large inductance L (several Henrys), a very small capacitance Cs (<0.5 pF) in series, and a series resistance r (a few hundred ohms), and a parallel capacity Cp (a few pF) which represents the electrostatic capacity between the two sides of the crystal.

The quality factor Q is high (> 20000). If “r” is neglected, the crystal impedance Z (jw) is given by:

Crystal impedance Z formula - Crystal Oscillator

Piezoelectric crystal symbol, equivalent circuit and reactance graphic - Crystal oscillators

The equation above indicates that the crystal has two resonant frequencies: a series resonance “ws” and a parallel resonance “wp” given by the following equations:

Crystal two resonant frequencies formulas

You may also like: Resonance in a RLC Circuits 

The last equation can be rewritten in the form:

reactance_resonance_frequency

where wp > ws since Cp > Cp.

A graphical representation of the reactance (impedance) Z (jw) is shown on picture above.

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