Quartz Crystal Operation & Theory

- quartz crystals work by converting electrical signals into mechanical vibrations within the crystal which are affected by the mechanical resonances of the physical crystal and then converted back to the electrical domain affecting the circuits accordingly.

The operation of quartz crystals depends upon the piezo-electric effect.

The theory shows that the electrical impulses 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 and show the very low levels of loss and as the theory shows this means that the selectivity or Q - quality factor is exceedingly high.

Piezo-electric effect

The peizo-electric effect is a key to the theory and operation of quartz crystals. It is an effect that occurs in a number of materials, both naturally occurring and synthesized. It obviously occurs in quartz and a number of ceramic materials used in the electronics industry as well as a number of organic substances. It can be used in a variety of ways within electronics and the electrical industry. Not only is it used in many resonant components, but it can also be used for electric transducers.

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 te material, or placing a charge across the material will cause a stress to be set up.

The piezo-electric effect is the key to the operation and theory behind quartz crystals.

Basic quartz crystal operation

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, all on different frequencies. 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. There are three main ways in which a crystal can vibrate: longitudinal mode, low frequency face shear mode, and high frequency shear. A cut known as the AT cut used for most crystals used in traditional radio and electronics circuits uses the high frequency shear mode.

Vibrational modes of a quartz crystal resonator
Vibrational modes of a quartz crystal resonator
(For the sake of clarity, the movements have been greatly exaggerated)

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.

The 4 parameter theoretical equivalent circuit of a quartz crystal showing the series resistance, capacitance and inductance together with a parallel capacitance
Quartz crystal theoretical 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.

    Quartz crystal series resonant frequency equation

    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.

    Quartz crystal parallel resonant frequency equation

    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.

Impedance characteristics of a quartz crystal resonator showing the two resonance points - one for series and one for parallel resonance
Impedance characteristics of a quartz crystal resonator

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.

Overtone operation

Apart from crystals operating in their fundamental mode, they can also operate in an overtone mode.

For an AT cut crystal, these overtone modes are nearly at the overtone number times the frequency of the fundamental. In fact the actual frequency of the overtone more nearly equates to the series mode fundamental frequency times the overtone number.

It should also be remembered that crystals may also have many other modes of operation that can be exercised. These modes are unwanted and may be excited to lesser or greater degrees by different circuits. Care should be taken especially when using crystals in unturned digital circuits as unwanted modes may unexpectedly dominate. Any tuning will naturally tend to suppress these unwanted modes.

The major applications for overtone crystals are for frequencies above 10 MHz and more. Here the crystals typically vibrate in a thickness shear mode and the the crystals can be excited in either fundamental or odd overtones. It is found that the motional capacitance C1n of an overtone crystal decreases. It follows an approximate law where the capacitance for the nth overtone is the capacitance for the fundamental divided by the square of the number of the overtone.

Pulling quartz crystals

In some instances it is necessary to be able to pull the resonant frequency of the crystal. This can be achieved by the external circuit when the crystal is operating in its parallel resonant mode. [Series mode is affected by a very much smaller degree].

A factor, often referred to as the crystal pulling factor can be calculated to determine the degree to which the frequency can be pulled or changed using external components.

Quartz crystal pulling factor equation - it indicates the sensitivity of the crystal to having its frequency 'pulled' by external components

This formula indicates the sensitivity of the crystal to changes in the external circuit. As a rough guide this factor may be around 200 for an AT cut crystal for use in its fundamental mode at frequencies above about 10 MHz.

The formula holds true for crystals operating in their fundamental mode. It is found the ratio for crystals operating in an overtone mode increases approximately as the square of the overtone number.

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.

Quartz crystal Q, quality factor equation

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.

By Ian Poole

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