Quartz Crystal Theory

Modes of vibration, cuts and frequency ranges

The AT-cut resonator uses the thickness shear mode of vibration (fig.1). A standing wave is set up in the crystal blank by the reflection at both major surfaces of traverse waves travelling in the thickness direction. The major mechanical displacement is in the plane of the crystal at right angles to the direction of wave propagation. At resonance on odd number of half wave lengths are contained in the thickness plane of the crystal blank. Therefore the thickness is the primary frequency determining dimension.

The AT-cuts (fig.2) are commonly manufactured in the frequency ranges:

1 MHz ~ 32 MHz as fundamental
30 MHz ~ 250 MHz as overlone (3rd; 5th; 7th; 9th)

Below about 1 MHz the thickness shear mode resonators become too cumbersome and unwieldy for general use and other modes of vibrations are used:

a) below about
100 kHz 		flexural, length extensional mode
b) 100 kHz face shear, CT; DT; SL cuts(fig.1)

For each mode of vibration there is an optimal angle of cut wich controls the frequency deviation of the quartz crystal over the temperature range.
Fig 1. Modes of Vibration


Fig 2. Angle of Cuts
Explanation of Crystal Units

The crystal unit is an electronic device that is capable of composing a frequency generator circuit with an extremely high stability. It has achieved high degree of compactness and high performance, along with the progress of other electronic components.

1 Equivalent Circuit
Vibration of a crystal unit is actually mechanical vibration. However, the crystal unit can be expressed by a two terminal network if its behavior is electrically converted. The series circuit consisting of L1, C1 and R1 is related to elastic vibration, while the element Co connected in parallel to the series arm as a capacitance attributable to the dielectric body of a quartz crystal plate. The resistance R1 is a resonance resistance of the crystal unit at the series resonance frequency. (See Fig. 3.)
Fig. 3 Equivalent Circuit of Crystal unit

2 Frequency-Temperature Characteristics

To use a crystal unit as an oscillator, its oscillated frequency is required to be stable against temperature variations. A quartz crystal has crystallographic axis, and crystal cut is defined according to the cutting angle against a crystallographic axis and its associated mode of vibration. Typical types of crystal cut and frequency-temperature characteristics are shown in Fig. 4.

Fig. 4 Frequency-Temperature Characteristics of Various Cutting Types
3  Frequency-Temperature Characteristics of an AT-Cut

The frequency-temperature characteristics of an AT-cut crystal unit most generally used at present are expressed by cubic curves. (See Fig. 5.) A crystal plate is cut at an angle at which a required frequency tolerance is obtained in the given operating temperature range. Actually, however, there can be some dispersion in apparent cutting angle due to the result of cutting and polishing accuracy in the successive processes. Therefore, it is necessary to raise processing accuracy.


Fig. 5 Frequency-Temperature Characteristics of an AT-Cut Crystal Unit