Capacitive Micromachined Ultrasonic Transducer Cells Research Articles

Design and Finite Element Simulation of a Novel 3D-CMUT Device for Simultaneous Sensing of In-Plane and Out-of-Plane Displacements of Ultrasonic Guided Waves.

In this study, we introduce a physical model of a three-dimensional (3D) guided wave sensor called 3D-CMUT, which is based on capacitive micro-machined ultrasonic transducers (CMUTs). This 3D-CMUT sensor is designed to effectively and simultaneously obtain 3D vibration information about ultrasonic guided waves in the out-of-plane (z-direction) and in-plane (x and y-directions). The basic unit of the 3D-CMUT is much smaller than the wavelength of the guided waves and consists of two orthogonal comb-like CMUT cells and one piston-type CMUT cell. These cells are used to sense displacement signals in the x, y, and z-directions. To ensure proper functioning of the 3D-CMUT unit, the resonant frequencies of the three composed cells are set to be identical by adjusting the microstructural parameters appropriately. Moreover, the same sensitivity in the x, y, and z-directions is theoretically achieved by tuning the amplification parameters in the external circuit. We establish a transient analysis model of the 3D-CMUT using COMSOL finite element simulation software to confirm its ability to sense multimode ultrasonic guided waves, including A0, S0, and SH0 modes. Additionally, we simulate the ball drop impact acoustic emission signal on a plate to demonstrate that the 3D-CMUT can not only utilize in-plane information for positioning but also out-of-plane information. The proposed 3D-CMUT holds significant potential for applications in the field of structural health monitoring (SHM).

New Insights for Parasitic Effects of Label-Free Biosensors Based on Capacitive Micromachined Ultrasonic Transducers

Capacitive micromachined ultrasonic transducers (CMUTs) are regarded as an attractive candidate in bio-applications such as imaging and molecule monitoring. However, the previous researches on biochemical sensing are mostly air-coupled application based on a CMUTs array because cell-to-cell mutual radiation and large motional loss in liquid environment are able to produce nonignorable noise. For a CMUTs cell, its characteristics (e.g., electrical, mechanical, and acoustic) are susceptible to parasitic effects owning to the physically capacitive dielectric dispersion, motional damping, and connection loss. Neglecting the parasitic effects on multidomain characteristics of CMUTs leads to significant robustness errors. Furthermore, finite element method (FEM) is not sufficient enough to model such parasitic effects when considering an integrated circuit interface and the evaluation of system responses. This article highlights a lumped element model (LEM) to analyze the behaviors of a label-free biosensor based on a single CMUTs cell directly operating in liquid. We successfully explore the performance of the biosensor with different types of parasitic effects in transient and frequency domains through LEM and FEM. The parasitic effects on mass loading of sensing layer and biomolecule such as deoxyribonucleic acid (DNA) and Immunoglobulin G (IgG) probes are predicted by LEM based on their surface mass densities. The proposed approach is capable of analyzing the variation’s margin of CMUTs-based biosensors to improve robustness for parasitic extraction in liquid.

Design and analysis of CMUT device using COMSOL for Radio frequency applications

This paper presents the design and analysis of Capacitive Micro-machined Ultrasonic Transducers (CMUTs) for a resonant frequency of 1.48 MHz. The CMUT has been designed using COMSOL and the results are found to be closely matching with the analytical calculations. The circular CMUT cell consists of polysilicon and silicon dioxide as the top membrane and polysilicon as the bottom membrane with an air gap. The eigen frequency analysis and pull-in analysis of the device has been performed. The solid mechanics module and electrostatic module of COMSOL Multiphysics have been used for performing the simulations. This device can be used for radio frequency applications as it has low displacement.

Use of Supporting Post Width to Increase the CMUT’s Resonant Frequency

A new method is proposed to increase the resonant frequency of a Capacitive Micromachined Ultrasonic Transducer (CMUT) device while keeping the radii unchanged for devices fabricated by sacrificial release (SR) method. The resonant frequency of a CMUT cell is determined by the properties of the membrane such as the Young’s modulus, membrane dimensions and effective membrane mass. Effective Young’s modulus is used to calculate the resonant frequency of CMUT fabricated by SR method. The supporting post structure will affect the boundary conditions at the edge of the CMUT membrane and in turn affect the deflection together with the effective Young’s modulus of the membrane. The perturbation method is used to derive the solution for the governing von Kármán equations. The results show that the thicker is the supporting post width, the higher is the resonant frequency. This method can also be used to simplify the mask design and CMUT fabrication process.

Investigation on Design Theory and Performance Analysis of Vacuum Capacitive Micromachined Ultrasonic Transducer.

The capacitive micromachined ultrasonic transducer (CMUT), as a new acoustic-electric conversion element, has a promising application prospect. In this paper, the structure of the vacuum capacitive micromachined ultrasonic transducer is presented, and its performance-influencing factors are investigated. Firstly, the influencing factors of the performance parameters of the vacuum CMUT are analyzed theoretically based on the circular plate model and flat plate capacitance model, and the design principles of the structural parameters of the CMUT cell are proposed. Then, the finite element simulation software COMSOL Multiphysics is used to construct CMUT cell models with different membrane materials, membrane shapes, membrane radius thicknesses, and cavity heights for simulation verification. The results show that both the membrane parameters and the cavity heights affect the performance parameters of the Vacuum CMUT. In order to improve the efficiency of the CMUT, materials with low bending stiffness should be selected, and the filling factor of the membrane should be increased. In order to achieve high-transmission sound pressure, a smaller radius thickness and a larger cavity height should be selected. To achieve high reception sensitivity, a larger membrane radius thickness and a smaller cavity height should be selected. In order to obtain high fractional bandwidth, a larger membrane radius thickness should be selected. The results of this paper provide a basis for the design of Vacuum CMUT cell structure.

Tunable Q matching networks for capacitive ultrasound transmitters

Airborne capacitive micromachined ultrasonic transducers (CMUTs) have predominantly large input capacitive reactance with small series radiation resistance. To maximize the acoustic power radiation at resonance we employ low cost LC matching networks between the low output impedance driving source and CMUT transducer. Conventional LC networks, including pi and T-network topologies are employed to provide a high-Q match. An important free parameter that controls the bandwidth of match in pi and T-networks is the center impedance. Since a simple pi-network can be formed from two basic L-sections, the center impedance of two L-sections provides control over the bandwidth of match. Hence these are tunable-Q matching networks. We use the lumped circuit model of CMUT to derive the input impedance of CMUT at resonance. At resonance the lumped equivalent circuit of transducer can be reduced to a series RC circuit consisting of radiation resistance and transducer’s equivalent capacitance only. From the input impedance of CMUT we determine the lumped inductance and capacitances of various LC matching networks. The aim of these matching networks is to cancel the large input capacitive reactance of CMUT cell and to match the cell’s radiation resistance to source resistance at resonance. We report significant improvement in the radiated power when CMUT is driven at resonance using the proposed matching schemes.

Design of a hexagonal air-coupled capacitive micromachined ultrasonic transducer for air parametric array

An air parametric array can generate a highly directional beam of audible sound in air, which has a wide range of applications in targeted audio delivery. Capacitive micromachined ultrasonic transducer (CMUTs) have great potential for air-coupled applications, mainly because of their low acoustic impedance. In this study, an air-coupled CMUT array is designed as an air parametric array. A hexagonal array is proposed to improve the directivity of the sound generated. A finite element model of the CMUT is established in COMSOL software to facilitate the choice of appropriate structural parameters of the CMUT cell. The CMUT array is then fabricated by a wafer bonding process with high consistency. The performances of the CMUT are tested to verify the accuracy of the finite element analysis. By optimizing the component parameters of the bias-T circuit used for driving the CMUT, DC and AC voltages can be effectively applied to the top and bottom electrodes of the CMUT to provide efficient ultrasound transmission. Finally, the prepared hexagonal array is successfully used to conduct preliminary experiments on its application as an air parametric array.

Fabrication and Characterization of a Wideband Low-Frequency CMUT Array for Air-Coupled Imaging

The capacitive micro-machined ultrasonic transducer (CMUT) has a good acoustic matching with air. In this paper, a 16-element CMUT array for airborne application is designed and fabricated by SOI wafer bonding process. All the CMUT cells have the same geometric parameters with circular and sealed membrane, which vibrate uniformly for good transmission efficiency. And the thin membrane with small mass is proposed to broaden the bandwidth of CMUT. The sensor array operates at a 215.6 kHz resonance frequency with a large −3dB fractional bandwidth of 15.7% and a good consistency. The selection of the driving parameters (i.e. the DC bias voltage and AC excitation signal) has been analyzed comprehensively, which can be used to optimize the driving strategy of the CMUT for better transmission and reception performance. The CMUT array proposed in this paper has been used to demonstrate a two-dimension non-contact air-coupled imaging by through-transmission mode and pitch-catch mode. The images of two kinds of objects have been reconstructed to exhibit the capability of the CMUT array in air-ultrasonic imaging, suggesting broad applications including 3D imaging, gesture recognition, and other human-machine interactions in air.

Experimental Characterization of an Embossed Capacitive Micromachined Ultrasonic Transducer Cell.

Capacitive Micromachined Ultrasonic Transducer (CMUT) is a promising ultrasonic transducer in medical diagnosis and therapeutic applications that demand a high output pressure. The concept of a CMUT with an annular embossed pattern on a membrane working in collapse mode is proposed to further improve the output pressure. To evaluate the performance of an embossed CMUT cell, both the embossed and uniform membrane CMUT cells were fabricated in the same die with a customized six-mask sacrificial release process. An annular nickel pattern with the dimension of 3 m × 2 m (width × height) was formed on a full top electrode CMUT to realize an embossed CMUT cell. Experimental characterization was carried out with optical, electrical, and acoustic instruments on the embossed and uniform CMUT cells. The embossed CMUT cell achieved 27.1% improvement of output pressure in comparison to the uniform CMUT cell biased at 170 V voltage. The fractional bandwidths of the embossed and uniform CMUT cells were 52.5% and 41.8%, respectively. It substantiated that the embossed pattern should be placed at the vibrating center of the membrane for achieving a higher output pressure. The experimental characterization indicated that the embossed CMUT cell has better operational performance than the uniform CMUT cell in collapse region.

Reception characteristics investigation and measurement of capacitive micromachined ultrasonic transducer

Purpose As a new type of ultrasonic transducer with significant advantages, capacitive micromachined ultrasonic transducer (CMUT) has good application prospect. The reception characteristic of the CMUT is one of the important factors determining the application effect. This paper aims to study the reception characteristics of CMUT. Design/methodology/approach In this paper, the state equation is deduced and the analysis model is established in SIMULINK environment based on the lumped parameter system model of the CMUT cell. Based on this analysis model, the influencing factors of CMUT reception characteristics are studied and investigated, and the time-domain and frequency-domain characteristics are investigated in detail. Findings The analysis results show that parameters directly affect the reception characteristics of the CMUT, such as direct current (DC) bias voltage, input sound pressure amplitude and frequency. At the same time, the measurement system is built and the reception characteristics are verified. Originality/value This paper provides an effective method for rapid analyzing the reception characteristics of CMUT. These results provide an important theoretical basis and reference for further optimization of CMUT structure design, and lay a good foundation for the practical application measurement.

A Nonlinear Lumped Equivalent Circuit Model for a Single Uncollapsed Square CMUT Cell.

An accurate nonlinear lumped equivalent circuit model is used for modeling of capacitive micromachined ultrasonic transducers (CMUTs). Finite-element analysis (FEA) is a powerful tool for the analysis of CMUT arrays with a small number of cells while with the harmonic balance (HB) analysis of the lumped equivalent circuit model, the entire behavior of a large-scale arbitrary CMUT array can be modeled in a very short time. Recently, an accurate nonlinear equivalent circuit model for uncollapsed single circular CMUT cells has been developed. However, the need for an accurate large-signal circuit model for CMUT cells with square membranes motivated us to produce a new nonlinear large-signal equivalent circuit model for uncollapsed CMUT cells. In this paper, using analytical calculations and FEA as the tuning tool, a precise large signal equivalent circuit model of square CMUT dynamics was developed and showed excellent agreement with finite-element modeling (FEM) results. Then, different CMUT single cells with square and circular membranes were fabricated using a standard sacrificial release process. Model predictions of resonance frequencies and displacements closely matched experimental vibrometer measurements. The framework presented here may prove valuable for future design and modeling of CMUT arrays with square membranes for ultrasound imaging and therapy applications.

An Analytical Model for Bandwidth Enhancement of Air-Coupled Unsealed Helmholtz Structural CMUTs

Capacitive micromachined ultrasonic transducers (CMUTs) were reported to own high potential in air-coupled ultrasonic applications such as noncontact nondestructive examination and gas flow measurement. The unsealed CMUTs which utilized the squeeze film effect were reported to overcome the narrow output pressure bandwidth of the conventional sealed CMUTs in air operation. This kind of unsealed CMUTs can also be regarded as Helmholtz resonators. In this work, we present the air-coupled unsealed Helmholtz structural CMUTs which utilize both the squeeze film effect and the Helmholtz resonant effect to enhance the output pressure bandwidth. Based on the mechanism of vibration coupling between membrane and air pistons in membrane holes, we propose an analytical model to aid the design process of this kind of CMUTs. We also use finite element method (FEM) to investigate this kind of CMUTs for our analytical model validation. The FEM results show that the significant bandwidth enhancement can be achieved when the Helmholtz resonant frequency is designed close to the fundamental resonant frequency of the CMUT membrane. Compared with the conventional sealed CMUT cell, the 4-hole unsealed Helmholtz structural CMUT cell improves both the 3-dB fractional bandwidth and SPL-bandwidth product around 35 times. Furthermore, it is found that, with more holes under the same hole area ratio or with a smaller ratio of the cavity height to the viscous boundary layer thickness, the Helmholtz resonant effect becomes weaker and thus the output pressure bandwidth decreases.

Sensitivity Optimization of Wafer Bonded Gravimetric CMUT Sensors

Optimization of the mass sensitivity of wafer bonded resonant gravimetric capacitive micromachined ultrasonic transducers (CMUTs) is presented. Gas phase sensors based on resonant gravimetric CMUTs have previously been demonstrated. An important figure of merit of these sensors is the sensitivity which, for typical CMUT geometries, is increased by decreasing the radius of the CMUT cell. This paper investigates how to minimize the radius of CMUT cells fabricated using the wafer bonding process. The design and process parameters affecting the radius of the CMUT and hereby the sensitivity are studied through numerical simulations and atomic force microscopy measurements. An excellent fit was obtained between the simulations and measured profiles with a low relative error of ≤ 5%, thus validating the simulation model. Two types of CMUTs are designed and fabricated using the design and process rules determined herein, with experimentally determined mass sensitivities of 0.46 Hz/ag and 0.44 Hz/ag, respectively. The two CMUT devices have cavities made using the local oxidation of silicon (LOCOS) and reactive ion etching (RIE) process. For the LOCOS process, it was found that the smallest radius can be obtained by choosing a Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> oxidation mask and lowering the pad SiO2 thickness, vacuum gap height, and Si bump height. For the RIE process, the vertical dimensions do not influence the horizontal dimensions and consequently, equivalent rules do not exist.

A Fast-Switching (1.35-μs) Low-Control-Voltage (2.5-V) MEMS T/R Switch Monolithically Integrated With a Capacitive Micromachined Ultrasonic Transducer (CMUT).

This paper describes the design and fabrication of an electrostatic MEMS switch that can be co-fabricated on the same substrate with a capacitive micromachined ultrasonic transducer (CMUT) as a transmit/receive (T/R) switch. The structure of the switch is modified from a single CMUT cell. An interrupted transmission line is defined across the center of the cell with control electrodes on both sides to pull a movable plate down. The plate has an insulation layer underneath and a metal bump is formed on the insulation layer and aligned to the transmission line gap, so that the switch could be turned on by pulling down the plate with electrostatic force and making the metal bump close the gap in the transmission line. The switch was designed using a finite-element model (FEM) and fabricated on a glass substrate using anodic bonding. A static characterization was first performed on a switch test structure, which shows the dc switching voltage was 68 V and the on-resistance was 50 Ω. The RF in and RF out isolation was measured as approximately 66 dB and insertion loss was approximately 4.5 dB for frequency range commonly used for medical ultrasound imaging. Then we performed the dynamic characterization in immersion. By setting the dc bias at 67 V, we found that the switch could be operated with a control-voltage as low as 2.5 V. The switching and release times are related to the rise time and the fall time of the control signal, respectively. The minimum switching time was measured as 1.34 μs with a control signal rise time of 300 ns, and the minimum release time was measured as 80 ns with a control signal fall time of 20 ns. We further demonstrated that a 1-kHz control signal with the optimized rise and fall times can be used to conduct and block a sinusoidal signal with 1-MHz frequency and 300-mV pp amplitude, as well as unipolar pulses with 5-V pp amplitude, 500-ns pulse width, and 2-kHz repetition rate. The presented MEMS switch could potentially eliminate the high-voltage process requirement for the on-chip front-end electronics of a CMUT-based ultrasound imaging system and thus improve the overall system efficiency.

Design and Fabrication of a High-Power Air-Coupled Capacitive Micromachined Ultrasonic Transducer Array With Concentric Annular Cells

One shortcoming of capacitive micromachined ultrasonic transducers (CMUTs) for commercial use is the weak output power relative to traditional ultrasonic transducers. Recently, we reported an annular CMUT cell with an improved transmit efficiency over a conventional circular cell in air. Extending this paper, we designed and fabricated a nine-element concentric CMUT array to enhance the transmit power and offer depth focusing. The proposed 200-kHz array has an aperture diameter of 2.13 cm and a fill factor of 81%. A pillar-free etching process was developed to create the deep large-area cavities of the proposed wafer-bonded CMUT. The fabricated device was characterized for the static and dynamic performance using a profilometer and a laser Doppler vibrometer, respectively. The average maximum plate dynamic displacements of the CMUT driven by 20-Vpp ac excitation voltage at dc biases of 100 and 150 V were measured to be 1.28 and $1.97~\mu \text{m}$ , respectively. Accordingly, the surface output power densities were calculated to be 0.40 and 0.96 KW/m2. We investigated the crosstalk between neighboring cells and the plate-cracking phenomenon, and provided relative suggestions for improvement. This paper demonstrates the feasibility of the concentric annular-cell CMUT array design for air-coupled applications.

An Optimization and Comparative Study of Air-Coupled CMUT Cells With Circular and Annular Geometries.

Air-coupled capacitive micromachined ultrasonic transducers (CMUTs) with annular cell geometry have recently been reported to have a promising transmit sensitivity. This paper reports three optimization schemes, which further improve the transmit sensitivity and also help achieve a reasonable comparison between the novel annular and conventional circular cells. Lumped element models of both cell types with laminate plate structures are presented. Based on these models, a design optimization flowchart was constructed to facilitate analytical optimization on the three schemes. Circular and annular CMUTs with a common 97-kHz natural resonance frequency were fabricated and characterized to verify the efficacy of the optimization principle. Using the optimization flowchart, annular and circular cells with frequencies ranging from 100 to 300 kHz were analytically optimized and then compared. The comparison results demonstrate that, given the same dc bias and ac excitation voltage, the output power density at the plate surface of the optimized annular cell is double that of the optimized circular cell. Additionally, when generating the same surface power density, an optimized annular cell requires either half the dc bias or half the ac excitation voltage of an optimized circular cell. This paper provides a practical optimization framework for CMUT cell design and demonstrates the superiority of annular cells for air-coupled applications.

Fabrication of Vacuum-Sealed Capacitive Micromachined Ultrasonic Transducers With Through-Glass-Via Interconnects Using Anodic Bonding

This paper presents a novel fabrication method for vacuum-sealed capacitive micromachined ultrasonic transducer (CMUT) arrays that are amenable to 3D integration. This paper demonstrates that MEMS structures can be directly built on a glass substrate with preformed through-glass-via (TGV) interconnects. The key feature of this new approach is the combination of copper through-glass interconnects with a vibrating silicon-plate structure suspended over a vacuum-sealed cavity by using anodic bonding. This method simplifies the overall fabrication process for CMUTs with through-wafer interconnects by eliminating the need for an insulating lining for vias or isolation trenches that are often employed for implementing through-wafer interconnects in silicon. Anodic bonding is a low-temperature bonding technique that tolerates high surface roughness. Fabrication of CMUTs on a glass substrate and use of copper-filled vias as interconnects reduce the parasitic interconnect capacitance and resistance, and improve device performance and reliability. A 16×16-element 2D CMUT array has been successfully fabricated. The fabricated device performs as the finite-element and equivalent circuit models predict. A TGV interconnect shows a 2-Ω parasitic resistance and a 20-fF shunt parasitic capacitance for 250-μm via pitch. A critical achievement presented in this paper is the sealing of the CMUT cavities in vacuum using a PECVD silicon nitride layer. By mechanically isolating the via structure from the active cells, vacuum sealing can be ensured even when hermetic sealing of the via is compromised. Vacuum sealing is confirmed by measuring the deflection of the edge-clamped thin plate of a CMUT cell under atmospheric pressure. The resonance frequency of an 8-cell 2D array element with 78-μm diameter circular cells and a 1.5-μm plate thickness is measured as 3.32 MHz at 15-V dc voltage (80% V pull-in ). [2016-0200].

Fabrication and characterization of a high frequency and high coupling coefficient CMUT array

Fabrication and initial measurement results of a bisbenzocyclobutene (BCB) based capacitive micromachined ultrasonic transducer (CMUT) linear phased array has been presented. The 32 channel 40 MHz CMUT array has been fabricated using a BCB based adhesive wafer bonding technique where BCB has been used as the diaphragm structural material, the bottom insulation layer, and also as the interelectrode insulator. The array has an aperture size of 0.62 mm. Each element consists of 264 square shaped cells. Each CMUT cell was designed to have a diaphragm sidelength of 10 μm with a cavity thickness of 0.75 μm. A low resistivity (0.01 Ω-cm) silicon wafer has been used as the bottom electrode. Series resonance of the array elements has been measured as 48 MHz which is within 20% agreement with the 3D FEA values simulated using IntelliSuite™, a MEMS design tool. Electromechanical coupling coefficient has been extracted to be 0.66 from the electrical input impedance measurement using an Agilent vector network analyzer with 20 V bias voltage. The measured static capacitance of individual cell is in excellent agreement with the 3-D FEA model with a deviation of 1.27%. The fabricated array can be used for anterior segment ophthalmic imaging, cardiovascular imaging, dermatology, small animal imaging, and biometric authentication. The technique can be used to fabricate high performance CMUT arrays for other applications.

A robust multi-objective and multi-physics optimization of multi-physics behavior of microstructure

A new strategy is presented to solve robust multi-physics multi-objective optimization problem known as improved multi-objective collaborative optimization (IMOCO) and its extension improved multi-objective robust collaborative (IMORCO). In this work, the proposed IMORCO approach combined the IMOCO method, the worst possible point (WPP) constraint cuts and the Genetic algorithm NSGA-II type as an optimizer in order to solve the robust optimization problem of multi-physics of microstructures with uncertainties. The optimization problem is hierarchically decomposed into two levels: a microstructure level, and a disciplines levels. For validation purposes, two examples were selected: a numerical example, and an engineering example of capacitive micro machined ultrasonic transducers (CMUT) type. The obtained results are compared with those obtained from robust non-distributed and distributed optimization approach, non-distributed multi-objective robust optimization (NDMORO) and multi-objective collaborative robust optimization (McRO), respectively. Results obtained from the application of the IMOCO approach to an optimization problem of a CMUT cell have reduced the CPU time by 44% ensuring a Pareto front close to the reference non-distributed multi-objective optimization (NDMO) approach (mahalanobis distance, DM2 =0.9503 and overall spread, So=0.2309). In addition, the consideration of robustness in IMORCO approach applied to a CMUT cell of optimization problem under interval uncertainty has reduced the CPU time by 23% keeping a robust Pareto front overlaps with that obtained by the robust NDMORO approach (DM2 =10.3869 and So=0.0537).

Lumped element modeling of air-coupled capacitive micromachined ultrasonic transducers with annular cell geometry

Air-coupled capacitive micromachined ultrasonic transducers (CMUTs) based on annular cell geometry have recently been reported. Finite element analysis and experimental studies have demonstrated their significant improvement in transmit efficiency compared with the conventional circular-cell CMUTs. Extending the previous work, this paper proposed a lumped element model of annular-cell CMUTs. Explicit expressions of the resonance frequency, modal vector, and static displacement of a clamped annular plate under uniform pressure were first derived based on the plate theory and curve fitting method. The lumped model of an annular CMUT cell was then developed by adopting the average displacement as the spatial variable. Using the proposed model, the ratio of average-to-maximum displacement was derived to be 8/15. Experimental and simulation studies on a fabricated annular CMUT cell verified the effectiveness of the lumped model. The proposed model provides an effective and efficient way to analyze and design air-coupled annular-cell CMUTs.