The Piezoelectric Resonator and the Effect of Electrode Spacing upon Frequency

Abstract

A new formulation of piezo‐resonator theory, called for convenience the ``mechanical theory,'' is developed for a crystal resonator with gap between crystal and electrodes, involving the increase in effective stiffness caused by space or surface charge, and explaining the increase in resonance frequency with increasing gap. The cases treated are thickness vibrations (both compressional and shear) of a plate, and lengthwise compressional vibrations of a bar. Specialized values of the elastic, dielectric and piezoelectric constants are employed, depending upon the nature of the elastic reactions for each cut and for each mode of vibration. The mechanical theory is compared with the customary ``electrical theory,'' according to which the resonator maintains a constant stiffness, the increase in frequency with increasing gap being accounted for by the electrical reactance of the gap considered as a series capacitance. When the crystal surfaces are equipotential (thickness vibrations, or lengthwise vibrations of a bar whose surfaces have been covered with a metallic deposit), the two theories are in agreement. It is shown that the mechanical theory, unlike the electrical theory, describes correctly the performance of a vibrating crystal bar that is not metallically coated. The characteristics of the resonator and the contrast between the two theories are illustrated by resonance‐circle and phase diagrams. The relative increase in frequency caused by the gap is shown to be Δf/f₀=Uw/(e+kw), where f₀ is the frequency at zero gap, f₀+Δf the frequency with a gap of width w, e the thickness of the crystal in the direction of the applied electric field, k the dielectric constant and U=16ε²/πc, ε being the appropriate piezoelectric constant and c the effective elastic constant at zero gap. In the case of the uncoated bar, the function U has to be multiplied by π²/8. Observations were made with eight quartz crystals of the following cuts: Y‐cut, X‐cut, BT‐ and AC‐cuts (notation of Lack, Willard and Fair) for thickness vibrations, and an X‐cut bar for lengthwise vibrations. With small gaps, the experimental values of the function U are, except with the BT‐ and AC‐cuts, from 13 to 60 percent smaller than the theoretical values. In most cases the discrepancy becomes still greater at large gap values. The probable reasons for these discrepancies are discussed, with the conclusion that a large increase in frequency with gap may be taken as a criterion of a good resonator (large piezoelectric constant, freedom from twinning and coupling). A Rochelle salt Y‐cut 45° bar when tested for the gap effect showed a satisfactory agreement with theory, considering the uncertainty in the values of the dielectric and piezoelectric constants. It is shown that the effect of gap upon frequency can be utilized in determining approximately the piezoelectric constant of a crystal. The theory of the dependence of frequency upon gap for overtone vibrations is briefly discussed.