A laminated plate theory for high frequency, piezoelectric thin-film resonators

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Abstract
A high frequency, piezoelectric, laminated plate theory is developed and presented for the purpose of modeling and analyzing piezoelectric thin‐film resonators and filters. The laminated plate equations are extensions of anisotropic composite plate theories to include piezoelectric effects and capabilities for modeling harmonic overtones of thickness‐shear vibrations. Two‐dimensional equations of motion for piezoelectric laminates were deduced from the three‐dimensional equations of linear piezoelectricity by expanding the mechanical displacements and electric potential in a series of trigonometric function, and obtaining stress resultants by integrating through the plate thickness. Relations for handling the mechanical and electrical effects of platings on the top and bottom surfaces of the laminate are derived. A new matrix method of correcting the cutoff frequencies is presented. This matrix method could also be used to efficiently correct the cutoff frequencies of any nth order plate laminate theories which employ Mindlin’s form of polynomial expansion of mechanical displacements and electric potential through the plate thickness. The first order laminated plate theory with correction factors for cutoff frequencies and slope of the flexural branch at large frequencies was applied to a 2‐layer, zinc oxide‐silicon strip, and a 3‐layer, zinc oxide‐zinc oxide‐silicon strip. Open circuit, dispersion relations were generated for a range of volume fractions and compared to the exact dispersion relations. Both the 2‐layer and 3‐layer strip show similar qualitative comparison: The present theory compares fairly well with the exact dispersion relation for real wave numbers and nonpropagating (imaginary) wave numbers which are smaller than 0.5i. The extensional branch, and thickness‐shear branches begin to deviate from the exact solution when the nondimensionalized frequency is greater than one. Consequently, to maintain accuracy of solutions when using the present first order laminated plate theory, one should limit the calculation of resonant, nondimensionalized frequencies to less than 1.1. Results for the frequency spectrum in the vicinity of the open circuit, fundamental thickness‐shear frequency, and modes shapes were presented for the 2‐layer strip with fixed edges, and a volume fraction of silicon equal to 0.2. The technically important fundamental thickness‐shear mode is found to have the shear component strongly coupled with extensional component.

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