How a Quartz Crystal Resonator Works

The operation of quartz crystals depends upon the piezo-electric effect that links mechanical resonances to the electrical system.

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The operation of quartz crystal resonators or xtals depends upon the piezo-electric effect and the sharp mechanical resonances of the material.

The electrical signals are converted from their electrical form into mechanical vibrations. It is these vibrations that are affected by the mechanical resonances of the quartz crystal and then linked back into electrical system.

The losses associated with quartz crystals are very low and this means that the selectivity or Q - quality factor is exceedingly high.

Quartz crystal resonator xtal circuit symbol — Circuit symbol for a quartz crystal resonator, xtal

Piezo-electric effect

When looking at how a quartz crystal works, it is first necessary to understand the basics of the piezo-electric effect.

It is an effect that occurs in a number of materials, both naturally occurring and synthetic.

The piezo electric effect occurs in quartz as well as a number of ceramic materials used in the electronics industry as well as a number of organic substances. It is for this reason that some ceramic resonators are available, although their performance does not equal that of quartz in terms of being used as a resonator.

The peizo-electric effect is not only used in resonators, but it can also be used for electric transducers - motion sensors, stress gauges, crystal microphones and many more.

When the piezo-electric effect occurs, an electric charge develops in certain solid materials as a result of an applied mechanical stress. This effect converts a mechanical stress in a crystal to a voltage and vice versa, i.e. a stress can cause a charge to be developed across the material, or placing a charge across the material will cause a stress to be set up.

Video: Understanding Quartz Crystal Resonators

How does a quartz crystal resonator work: the basics

For the operation of a quartz crystal it is found that the piezo-electric effect converts the electrical impulses to mechanical stress which is subject to the very high Q mechanical resonances of the crystal, and this is in turn linked back into the electrical circuit.

The quartz crystal can vibrate in several different ways, and this means that it has several resonances - each on a different frequency.

Fortunately the way in which the quartz crystal blank is cut from the original crystal itself can very significantly reduce this. In fact the angle of the faces relative to the original crystal axes determines many of its properties from the way it vibrates to its activity, Q, and its temperature co-efficient.

When looking at how a quartz crystal resonator works, it can be seen that there are three main ways in which a crystal can vibrate:

Longitudinal mode: In the longitudinal mode the crystal extends and shortens. This causes the centre to become thinner as the crystal extends and thicker as it shortens.

Quartz crystal resonator longitudinal vibration
Low frequency face shear mode: For crystals that work in this mode, the whole rectangular shape of the crystal vibrates as shown below.

Quartz crystal resonator low frequency shear vibration
High frequency shear: The high frequency shear mode is used for most quartz crystal resonators working in the region 1 MHz upwards.

Quartz crystal resonator high frequency shear vibration

The way in which the crystal blank is cut from the bulk crystal affects the way in which it vibrates. In particular the angle it is cut relative to the axes of the bulk quartz crystal has a major effect. There are several standard ‘cuts’ which are given names, and these have known and useful performance parameters. A cut known as the AT cut is used for most crystals found in traditional radio and electronics circuits and this typically vibrates using the high frequency shear mode.

Equivalent circuit of a quartz crystal resonator

To analyse the electrical response of a quartz crystal resonator, it is very often useful to depict it as the equivalent electrical components that would be needed to replace it. This equivalent circuit is can then be used to analyse its response and predict its performance as in the diagram below:

The equivalent circuit given below is often called the 4-parameter crystal model and it is sufficient for many calculations and to illustrate the operation of the crystal.

Quartz crystal resonator equivalent circuit

It is possible to equate these theoretical constituent components to real physical attributes of the crystal:

L: The inductance arises from the mass of the material.
C1: This capacitance arises from the compliance of the crystal.
R: This element arises from the losses in the system. The largest of these arises from the frictional losses of the mechanical vibration of the crystal.
Co : This capacitance in the theoretical quartz crystal equivalent circuit arises from the capacitance between the electrodes of the crystal element. This is often refered to as the shunt capacitance.

Apart from their use in oscillators, quartz crystals find uses in filters. Here they offer levels of performance that cannot be achieved by other forms of filter. Often several crystals may be used in one filter to provide the correct shape.

Crystal parallel and series resonance

There are two modes in which a crystal oscillator can operate and these can be seen from the equivalent circuit diagram.

Series resonance: This is a standard series resonance condition formed by the series connection of a capacitor and inductor. At the resonant frequency, fs, the capacitive and inductive reactances cancel and the impedance falls to a minimum equal to the resistance in the circuit, i.e. R.

$f_{s} = \frac{1}{2 π \sqrt{L C_{1}}}$

It is found that in this mode the external circuit has very little effect on the crystal resonance.
Parallel resonance: The parallel resonance for the quartz crystal condition is formed by a capacitor and inductor in parallel. At resonance the impedance of this circuit rises to a maximum. The actual resonant frequency, fp, derivation for this mode incorporates the inductance along with both capacitors seen in the equivalent circuit.

$f_{s} = \frac{1}{2 π \sqrt{L \frac{C_{o} C_{1}}{C_{o} + C_{1}}}}$

When operating in this mode it will be seen that any capacitance introduced by the external circuit will also have an effect. As a result this 'load' capacitance forms part of the specification for the operation of the crystal - load values of 30pF, 20pF, etc are seen in the specifications. The calculations in the design of the oscillator should enable the correct load capacitance to be provided to the crystal. Changing the load capacitance can also be used to trim the exact frequency of the crystal to ensure that it is on exactly the correct frequency. Typically a small variable trimmer capacitor may be added to ensure that the correct load capacitance is provided.

This mode is sometimes referred to as the crystal's anti-resonant frequency. The reason for this is that the impedance of the circuit reaches a peak at resonance.

Quartz crystal resonators can operate in either mode, and in fact the difference between the parallel and series resonant frequencies is quite small. Typically they are only about 1% apart.

Quartz crystal resonator impedance plot — Quartz crystal resonator impedance characteristic

Of the two modes, the parallel mode is more commonly used, but either may be used. Oscillator circuits for using the different modes are naturally different, as one oscillates when the crystal reaches its maximum impedance whilst the other operates when the crystal reaches its minimum impedance.

Crystal Q, quality factor

The Q or quality factor is an important aspect of quartz crystal resonance. Crystals offer very high levels of Q, sometimes in excess of 100 000.

Accordingly it is necessary to be able to calculate the level of Q to be able to determine other constraints and design considerations for the circuit in which it is to operate.

A straightforward equation is available to be able to calculate the value of Q for a given crystal.

Q = \frac{1}{2 π F_{s} R C_{1}}

From this it can be seen that the series capacitance has a major effect on the Q. Lowering the series capacitance increases the Q in direct proportion for a given frequency.

Quartz crystal resonators are a complex linkage between the electrical md mechanical domains. Although in theory it appears that theory operation appears quite straightforward, in practice many effects work together and they may link in unexpected manners. It is not unknown for quartz crystal resonators to be excited by a different mode in a particular circuit and hence operate on an incorrect frequency. Understanding how quartz crystal resonators work can help identify these issues in the unlikely event they may occur. Often adding a broadly resonant circuit to the oscillator to ensure that it can only operate in the required mode suffices in this instance.