Quartz Resonator & Oscillator Tutorial


	The oscillator designer treats the crystal unit as a circuit component and just deals with the crystal unit's equivalent circuit. Shown above is a simple equivalent circuit of a single-mode quartz resonator. A resonator is a mechanically vibrating system that is linked, via the piezoelectric effect, to the electrical world. A load capacitor C_L is shown in series with the crystal. C₀, called the "shunt" capacitance, is the capacitance due to the electrodes on the crystal plate plus the stray capacitances due to the crystal enclosure. The R₁, L₁, C₁ portion of the circuit is the "motional arm" which arises from the mechanical vibrations of the crystal.
	The C₀ to C₁ ratio is a measure of the interconversion between electrical and mechanical energy stored in the crystal, i.e., of the piezoelectric coupling factor, k. C₀/C₁ increases with the square of the overtone number; the relationship of C₀/C₁ to k and N is 2C₀/C₁ = [pN/2k]², where N is the overtone number. When a dc voltage is applied to the electrodes of a resonator, the capacitance ratio C₀/C₁ is also a measure of the ratio of electrical energy stored in the capacitor formed by the electrodes to the energy stored elastically in the crystal due to the lattice strains produced by the piezoelectric effect. The C₀/C₁ is also inversely proportional to the antiresonance-resonance frequency separation (i.e., the pole-zero spacing - see the page after next) which is an especially important parameter in filter applications. The slope of the reactance vs. frequency curve near f_s is inversely proportional to C₁, i.e., DX/(Df/f) ~ 1/pfC₁ near f_s, where X is the reactance. C₁ is, therefore, a measure of the crystal's "stiffness," i.e., its tunability. For a simple RLC circuit, the width of the resonance curve is inversely proportional to the quality factor Q, but in a crystal oscillator, the situation is complicated by the presence of C₀ and by the fact that the operating Q is lower than the resonator Q. For a quartz resonator, Q = (2pf_sC₁R₁)^-1.
	When the load capacitor is connected in series with the crystal, the frequency of operation of the oscillator is increased by a Df, where Df is given by the equation below the equivalent circuit. A variable load capacitor can thus be used to vary the frequency of the resonator-capacitor combination, which may be applied in, e.g., a VCXO or TCXO.
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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.

	V. E. Bottom, Introduction to Quartz Crystal Unit Design, Chapters 6 and 7, Van Nostrand Reinhold Company, 1982.