Radiation Impedance of Rectangular CMUTs

TL;DR

This study provides a fast, physics-informed method to estimate the acoustic radiation impedance of rectangular CMUT membranes by approximating the fundamental-mode velocity with a polynomial profile. Building on a Green's-function–based impedance integral, it yields tractable expressions for radiation resistance $R_R$ and reactance $X_R$, with a 1D variant improving accuracy for very long membranes. Validation against FEM across aspect ratios from 1:1 to 1:25 shows close agreement for $R_R$ and reasonable agreement for $X_R$ below $ka \approx 5$, while offering runs that are orders of magnitude faster than full FEM. The work enables rapid design exploration of rectangular CMUTs and includes a MATLAB script to facilitate impedance calculations, supporting higher-throughput optimization of these devices.

Abstract

Recently, capacitive micromachined ultrasound transducers (CMUTs) with long rectangular membranes have demonstrated performance advantages over conventional piezoelectric transducers; however, modeling these CMUT geometries has been limited to computationally burdensome numerical methods. Improved fast modeling methods such as equivalent circuit models could help achieve designs with even better performance. The primary obstacle in developing such methods is the lack of tractable methods for computing the radiation impedance of clamped rectangular radiators. This paper presents a method which approximates the velocity profile using a polynomial shape model to rapidly and accurately estimate radiation impedance. The validity of the approximate velocity profile and corresponding radiation impedance calculation was assessed using finite element simulations for a variety of membrane aspect ratios and bias voltages. Our method was evaluated for rectangular radiators with width:length ratios from 1:1 up to 1:25. At all aspect ratios, the radiation resistance was closely modeled. However, when calculating the radiation reactance, our initial approach was only accurate for low aspect ratios. This motivated us to consider an alternative shape model for high aspect ratios, which was more accurate when compared with FEM. To facilitate development of future rectangular CMUTs, we provide a MATLAB script which quickly calculates radiation impedance using both methods.

Figures (10)

• Figure S1: The approximate velocity profile of a rectangular CMUT with an aspect ratio of 4 is plotted using the proposed expression of \ref{['v(x,y)']}.
• Figure S2: The velocity profile of a rectangular 1:4 CMUT is depicted at the top while the static deflection is shown in the bottom half of the figure. For the velocity profile, the DC bias was set to 20V and the AC driving amplitude to 1V. The static deflection profile is depicted for the DC bias of 50V. These results were extracted from FEM simulations discussed in Section \ref{['FEMVal']}. Note the significant difference between the profile shapes.
• Figure S3: This figure shows the rectangular 1:10 CMUT simulation with COMSOL Multiphysics. A quarter of the space is simulated to reduce computational cost. The boxed corner is equivalent to the center of the CMUT.
• Figure S4: The cross-sectional velocity profiles from FEM and the proposed approximation of \ref{['v(x,y)']} for the square, 1:4, 1:10, and 1:25 rectangular CMUTs. The FEM velocity profiles were extracted from frequencies near the CMUT's fundamental resonance at the given DC bias. For the rectangular CMUTs, their length was along the y-axis. Note the shape change of the cross-sectional velocity profile along the length as the aspect ratio grows from 4 to 25. The absolute relative errors of the approximate profile compared to the FEM are stated in Table \ref{['tab_ARE']}.
• Figure S5: The radiation impedance of a circular CMUT obtained from simulation compared to the expression derived by M. Greenspan martin_greenspan. The error of the theoretical predictions compared to FEM are quantified in Table \ref{['tab_MAPE']}.