Trapped‐Energy Modes in Quartz Filter Crystals

The behavior of thickness/shear‐mode quartz resonators is explained both mathematically and experimentally in terms of lateral standing‐wave trapped‐energy modes. A cutoff phenomenon, similar to that occurring in optical total internal reflection, results in energy trapping or the restriction of vibratory energy almost exclusively to the electroded region of a resonator, with an energy distribution decreasing exponentially with distance from the electrode edge. Consideration of boundary conditions at the electrode edges of an idealized two‐dimensional model yields an expression for the eigenfrequencies of symmetric inharmonic overtone modes (lateral standing waves) that can exist in the electroded region of fundamental‐ and harmonic‐mode resonators. Design formulas for the suppression of these unwanted modes are included. They show that electrode thickness and lateral dimensions can be traded off against one another to obtain a wide range of motional parameters. Previously, unwanted responses were elimina...