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At frequencies far from the resonance
frequency, the resonator is a simple parallel plate capacitor having a
capacitance Co ko(A/t), where A is the area of the electrodes, t
is the thickness of the plate, k is the dielectric constant and o is the
permittivity of free space. The reactance is zero at resonance, and it is
maximum at the antiresonance frequency. The antiresonance to resonance
separation, fa - fr fr/2r’ – r’/2Q2, where r’ = Co/C1.
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In most oscillator circuits, the frequency
is in the region shown, where the resonator’s reactance is inductive. An adjustable capacitance in series (or
parallel) with the resonator can then used to adjust the frequency of
oscillation.
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The reactance vs. fractional frequency
slope, X/(f/f), is a measure of the resonator’s “stiffness,” i.e., the amount
the resonator’s frequency changes with load capacitance. The stiffer the resonator, the less the
resonator’s frequency changes with a change in load capacitance. Near fS, X/(f/f) = 1/(fSC1). Overtone resonators are stiffer than
fundamental mode units because the C1 of overtone resonators is
smaller than the C1 of fundamental mode units.
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A. Ballato,
"Piezoelectric Resonators," in B. Parzen, Design of Crystal and
Other Harmonic Oscillators, pp. 66-122, John Wiley and Sons, Inc., 1983.
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E. Hafner,
"Resonator and Device Measurements," in E. A. Gerber and A.
Ballato, Precision Frequency Control, Vol. 2, pp.1-44, Academic Press, 1985.
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V. E. Bottom, Introduction
to Quartz Crystal Unit Design, Chapters 6 and 7, Van Nostrand Reinhold
Company, 1982.
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