This question was previously asked in

ESE Electrical 2013 Paper 1: Official Paper

Option 4 : the inverse of the square root of the mass of the crystal

CT 3: Building Materials

2962

10 Questions
20 Marks
12 Mins

**Crystal oscillators** operate on the principle of inverse piezoelectric effect in which an alternating voltage applied across the crystal surfaces causes it to vibrate at its natural frequency. These vibrations which eventually get converted into oscillations.

__Piezoelectric effect in Quartz crystal:__

- It has the characteristic that when mechanical stress (compression or stretching) is applied to its two faces, a potential difference V is generated across it.
- On the other hand, when an electric potential difference is applied across the quartz crystal, the latter undergoes deformation. As soon as there is no voltage across it, the quartz crystal regains its initial shape.
- These properties are known as the piezoelectric effect.
- Owing to the piezoelectric effect, the quartz crystal behaves as a resonant RLC circuit.

**Electrical equivalent circuit:**

In crystal oscillators, the crystal is suitably cut and mounted between two metallic plates as shown by figure whose electrical equivalent is also shown in the figure.

**Mechanical equivalent circuit:**

The mechanical model of a crystal oscillator is a simple compliance (spring)—inertia (mass)—damping (dashpot) system.

**Oscillation frequency:**

Due to the presence of Cp, the crystal will resonate at two different frequencies:

The frequency of oscillations in the case of series resonance is,

\({\omega _s} = \frac{1}{{\sqrt {{L_s}{C_s}} }}\)

The frequency of oscillations in the case of parallel resonance is,

\({\omega _p} = \frac{1}{{\sqrt {{L_s}\frac{{{C_s}{C_p}}}{{{C_s} + {C_p}}}} }}\)

Because the electrical and mechanical models are assumed equivalent, the natural frequency of the mechanical system must equal the natural frequency of the electrical system. This yields:

\({\omega _s} = \sqrt {\frac{K}{M}} \)

K is the spring modulus and M is the mass of the crystal

From the above expression, it is clear that the tuning frequency is linearly proportional to the inverse of the square root of the mass of the crystal.