Electrostatic and Small-Signal Analysis of CMUTs With Circular and Square Anisotropic Plates

Traditionally, capacitive micromachined ultrasonic transducers (CMUTs) are modeled using the isotropic plate equation, and this leads to deviations between analytical calculations and finite element modeling (FEM). In this paper, the deflection is calculated for both circular and square plates using the full anisotropic plate equation. It is shown that the anisotropic calculations match excellently with FEM, whereas an isotropic approach causes up to 10% deviations in deflection. For circular plates, an exact solution can be found. For square plates using the Galerkin method, and utilizing the symmetry of the silicon crystal, a compact and accurate expression for the deflection can be obtained. The deviation from FEM in center deflection is <0.1%. The theory of multilayer plates is also applied to the CMUT. The deflection of a square plate was measured on fabricated CMUTs using a white light interferometer. Fitting the plate parameter for the anisotropic calculated deflection to the measurement, a deviation of 0.07% is seen. Electrostatic and small-signal dynamic analysis are performed using energy considerations including anisotropy. The stable position, effective spring constant, pullin distance, and pull-in voltage are found for both circular and square anisotropic plates, and the pressure dependence is included by comparison with the corresponding analysis for a parallel plate. Measurements on fabricated devices with both circular and square plates subjected to increasing bias voltage are performed, and it is observed that the models including anisotropic effects are within the uncertainty interval of the measurements. Finally, a lumped element small-signal model for both circular and square anisotropic plates is derived to describe the dynamics of the CMUT.