Quartz crystals
The importance of quartz crystal
resonators in electronics results from their extremely high Q,
relatively small size and excellent temperature stability.
A quartz crystal resonator uses the
piezoelectric properties of quartz. The direct piezoelectric effect
refers to the electric polarization of certain materials brought about
by the application of mechanical stress. The converse effect refers to
the deformation produced in the same materials by the application of an
electric field. In a quartz crystal resonator, a thin slice of quartz,
cut at an appropriate orientation with respect to the crystallographic
axes, is placed between two electrodes. An alternating voltage applied
to these electrodes causes the quartz to vibrate in sympathy. The
accompanying changes in the electric polarization constitute an
electric displacement current through the resonator.
When the frequency of the applied
voltage approaches one of the mechanical resonance frequencies of the
quartz slice, the amplitude of the vibrations becomes very large. The
accompanying displacement current also increases, so that the effective
impedance of the device decreases in magnitude. The rapid change in
impedance as the frequency varies in the neighbourhood of resonance is
the key factor in the application of quartz crystal resonators as
frequency control elements in crystal oscillators.
Electrically, a quartz crystal can be
represented by the equivalent circuit of Figure 1, where the series
combination R1, L1 and C1 represent the contributions to the impedance
from the piezoelectric effect, and C0 represents the shunt capacitance
between the electrodes along with any stray holder capacitances. The
inductance L1 is a function of the mass of the quartz while the
capacitance C1 is associated with its stiffness. The resistance R1
results from the loss in the quartz and in the mounting arrangement.
The parameters of the equivalent circuit can be measured to accuracies
of the order of 1%.
A reactance-frequency plot of the
equivalent circuit is given in Figure 2. There are many related
formulae for crystal
performance; the first of these is for fs. This is the frequency at
which the crystal is series resonant and is given by:
where fs is in Hz, L1 in henries and
C1 in farads.
Calibration tolerance
Calibration tolerance is the maximum
allowable deviation in frequency of a crystal at a specific
temperature, the reference temperature (usually 25°C).
Frequency stability
Crystals suffer instability from
several causes. Temperature variation and a physical change of mass
which results in the long-term drift we call ageing are probably those
which concern us most.
The effects of temperature variation
are minimized by an appropriate choice of crystal cut and (for close
tolerance requirements) by including a temperature dependent reactance
in the crystal’s circuit, or by holding it at a constant
temperature in a small oven. AT-cut crystals are the most widely used
today because their family of frequency-temperature curves readily
provides good performance at low cost for all but the most demanding
applications.
Uncompensated AT-cut crystals can be
specified with tolerances down to ±5 ppm from –10
°C to 60 °C, with larger tolerances required for wider
temperature ranges as illustrated in Figure 3, showing a typical family
of AT-cut frequency-temperature curves. These curves may be represented
by cubic equations and are strongly dependent on the angle of cut of
the quartz blank. The points of zero temperature coefficient are called
the upper and lower turning points. One turning point can be placed
where desired by selecting the angle of cut; the other is then fixed,
since both are symmetrical about a point in the
20°–30°C range. The slope between the turning
points becomes smaller as they move together. Crystals designed for use
in an oven are cut so that the upper turning point coincides with the
oven operating temperature.
Figure 4 shows the frequency
temperature curves from several low-frequency cuts. The J-cut is used
below 10 kHz, while an XY-cut may be used from 3 kHz to 85 kHz. An
NT-cut may be used in the 10 kHz range. A DT-cut is applicable from 100
kHz to about 800 kHz and a CT-cut from 300 kHz to 900 kHz.
Load capacitance
Crystals can be calibrated by their
manufacturer at either fr, where they appear resistive (or fs which is
very close to fr), or for resonance with a capacitive load, where of
course they must appear inductive. The latter condition is called load
resonance and represented in general terms by the symbol fL; more
specifically, the symbol f30 would, for example, represent the
frequency at which the crystal is at resonance with a 30 pF
capacitative load.
The point on the crystal’s
reactance curve at which calibration is needed is determined by the
circuit configuration. As a general rule, a non-inverting maintaining
amplifier in an oscillator requires calibration at fr and an inverting
amplifier needs calibration at some value of ‘load
capacitance’, CL. The latter arrangement relies upon the
inductive crystal, together with the load capacitance with which it is
at resonance, to provide a further 180° of phase shift.
The most common exception to the rule
is when a small capacitor, a varicap diode for example, is placed in
series with the crystal in the non-inverting amplifier circuit to
provide a degree of frequency adjustment. In such a case the crystal
must be calibrated for resonance with the mean value of that
capacitance.
Pullability
The pullability of a crystal is a
measure of its frequency change for a given change of load capacitance.
This is often expressed as the difference between its series resonance
frequency (fr) and its load resonance frequency (fL). This offset can
be calculated in parts per million usin fractional load resonance
frequency offset (DL), the actual frequency change from fr to fL for a
given value of CL.
where C1, C0 and CL are all expressed
in the same units.
Figure 5 shows a typical curve for the
effect of frequency change with respect to change in load capacitance.
Alternatively, it is common to express
a crystal’s pullability as a trim sensitivity in ppm per pF
change of load capacitance. This is given in ppm/pF by:
where C1, C0 and CL are in pF, and is
shown graphically in Figure 6 for various values of (C0 + CL).
Related links:
Application
note: Timekeeping with quartz crystals
Application
note: Jitter
Quartz
crystals
Frequency
control technologies
Markets
& applications - Frequency Control Products
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