Analytic solution for N-electrode actuated piezoelectric disk with application to piezoelectric micromachined ultrasonic transducers

In this work, the deflection equation of a piezoelectrically-driven micromachined ultrasonic transducer (PMUT) is analytically determined using a Green´s function approach. With the Green´s function solution technique, the deflection of a circular plate with an arbitrary circular/ring electrode geometry is explicitly solved for axisymmetric vibration modes. For a PMUT with one center electrode covering ≈60% of the plate radius, the Green´s function solution compares well with existing piece-wise and energy-based solutions with errors of less than 1%. The Green¿s function solution is also simpler than them requiring no numerical integration, and applies to any number of axisymmetric electrode geometries. Experimentally measured static deflection data collected from a fabricated piezoelectric micro ultrasonic transducer (PMUT) is further used to validate the Green´s function model analysis. The center deflection and deflection profile data agree well with the Green´s function solution over a range of applied bias voltages (5 to 21 V) with the average error between the experimental and Green´s function data less than 9%.