Introduction to quartz technology
Quartz Crystals, Cut Angles
Piezoelectric
materials, especially quartz , have the property to transform
electrical energy into mechanical energy and vice versa. In technical
applications this effect is utilised by applying an alternating
electrical field, which will cause the material to vibrate and
subsequently resonate mechanically. This electrical reaction permits
usage as an electrical resonator with a very high figure of merit Q and a
low temperature coefficient.
Different
quartz crystal cuts can be made possessing different properties. Cuts
are defined by two rotation angles phi and theta around the
crystallographic axes. Most common cuts are the single rotation AT- cut (
phi = 0° ) and the double rotation SC-cut ( phi = 22° ) . The theta
angle in both cases is around 34°. Other double rotated cuts like MSC-,
IT-, FC-, LD- for special applications also exist.
Quartz Resonators
The
active component of the crystal resonator is a mechanically vibrating
plate ( "crystal element" ) cut from mono-crystalline quartz with a
precise orientation to the crystallographic axes. The resonator is
plated under high vacuum with aluminium, silver or gold electrodes and
hermetically sealed into a suitable enclosure either with a coldweld or
resistance weld process . The physical dimensions of the element and its
orientation to the axes will determine in particular the resonance
frequency, its initial accuracy, its electrical properties and the
temperature coefficient. KVG produces AT- and SC-Cut crystals (and
others), which are the most widely used cuts providing a frequency range
from 800kHz up to 300MHz and excellent frequency-temperature
characteristics.
The
frequency of crystals is inversely proportional to the thickness of the
element . For mechanical processing, this results in an upper frequency
limit of about 50MHz for crystals operating on the fundamental mode. To
reach higher frequencies in the fundamental mode KVG also produces
chemically etched invertedmesa crystals where the central part of the
resonator is etched to have a thickness of as low as ten microns.
Resonator Design
Many
different parameters have an influence on the final resonator
properties. Thickness and diameter of the element , electrode diameter,
electrode material but also holders, sealing method etc. Crystal
elements can be manufactured plano-parallel or contoured ( with bevels,
plano-convex or bi-convex ) . Contouring is necessary to prevent edge
effects . A radius of curvature can be manufactured on one or both sides
of the crystal element to trap the energy in the center of the
resonator. Trapping can also be performed through mass loading.
Fundamental mode and overtone mode
High
frequency crystals vibrate in the thicknessshear vibration, which can
be excited in fundamental or odd overtone modes. The motional
capacitance C1n of an overtonecrystal decreases with the order n of the
overtone and is approximately given by C n C n 1 11 » 2 . Therefore the
ratio CO/C1 is much larger for overtone crystals than for crystals
operating in fundamental mode and the pulling range is reduced by a
factor of approximately n3. Crystals used in VCXOs, where wide pulling
range is required, therefore operate in fundamental mode.
Unwanted Response and Inharmonics ( spurious modes )
All crystal resonators produce for each overtone a main mode which is a thickness shear vibration and also unwanted responses, which are inharmonic thickness shear modes above the resonance frequency. Besides the commonly used thickness shear C-mode another thickness shear mode named B-mode exsists. It has a higher frequency and commonly lower motional resistance than the C-mode but a larger temperature coefficient . Sometimes it becomes necessary to filter this mode for the oscillator to work on the C-mode. Further unwanted modes are shear-, flexure-, thickness- and twist vibrations, which can appear above and below the required resonance frequency. With correct oscillator design the unwanted modes rarely cause problems. Unwanted modes close to the resonance frequency affect the start up behaviour of oscillators, or cause shifting to the wrong frequency during operation. Other undesired effects are frequency and resistance dips over temperature caused by unwanted modes. Spurious modes are generally specified as the ratio of resonance resistance of the inharmonic modes to the main mode resistance. KVG must have detailed information about the test circuit (e.g. pi-network or measurement bridge) and about the frequency range of the spurious modes.
Equivalent electrical circuit
Near to the resonance frequency the crystal unit is represented by an electrical two pole shown in figure8.
CO: shunt capacitance (capacitance between the electrodes, crystal holder, leads and case)
C1: motional capacitance (represent mechanical elasticity)
L1: motional inductance (represent mechanical inertia)
R1: motional resistance (represents mechanical losses)
Figure 9 shows the response in amplitude and phase vs frequency around resonance. The resonance frequency is given by:
