Capacitive micromachined ultrasonic transducers (CMUTs) are usually composed of large arrays of closely packed cells. In this work, we use an equivalent circuit model to analyze CMUT arrays with multiple cells. We study the effects of mutual acoustic interactions through the immersion medium caused by the pressure field generated by each cell acting upon the others. To do this, all the cells in the array are coupled through a radiation impedance matrix at their acoustic terminals. An accurate approximation for the mutual radiation impedance is defined between two circular cells, which can be used in large arrays to reduce computational complexity. Hence, a performance analysis of CMUT arrays can be accurately done with a circuit simulator. By using the proposed model, one can very rapidly obtain the linear frequency and nonlinear transient responses of arrays with an arbitrary number of CMUT cells. We performed several finite element method (FEM) simulations for arrays with small numbers of cells and showed that the results are very similar to those obtained by the equivalent circuit model.

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Equivalent circuit-based analysis of CMUT cell dynamics in arrays

A large-signal, transient model has been developed to predict the output characteristics of a CMUT array operated in the non-collapse mode. The model is based on separation of the nonlinear electrostatic voltage-to-force relation and the linear acoustic array response. For modeling of linear acoustic radiation and crosstalk effects, the boundary element method is used. The stiffness matrix in the vibroacoustics calculations is obtained using static finite element analysis of a single membrane which can have arbitrary geometry and boundary conditions. A lumped modeling approach is used to reduce the order of the system for modeling the transient nonlinear electrostatic actuation. To accurately capture the dynamics of the non-uniform electrostatic force distribution over the CMUT electrode during large deflections, the membrane electrode is divided into patches shaped to match higher order membrane modes, each introducing a variable to the system model. This reduced order nonlinear lumped model is solved in the time domain using commercial software. The model has two linear blocks to calculate the displacement profile of the electrode patches and the output pressure for a given force distribution over the array. The force-to-array-displacement block uses the linear acoustic model, and the Rayleigh integral is evaluated to calculate the pressure at any field point. Using the model, the time-domain transmitted pressure can be simulated for different large drive signal configurations. The acoustic model is verified by comparison to harmonic FEA in vacuum and fluid for high- and low-aspect-ratio membranes as well as mass-loaded membranes. The overall software model is verified by comparison to transient 3-D finite element analysis and experimental results for different large drive signals, and an example for a phased array simulation is given.

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A large-signal model for CMUT arrays with arbitrary membrane geometry operating in non-collapsed mode

Equivalent circuit models of large arrays of curved (spherical shape) and flat piezoelectric micromachined ultrasonic transducers (pMUTs) have been developed for complex pMUT arrays design and analysis. The exact solutions for circuit parameters in the electromechanical domain, such as mechanical admittance, input electrical impedance, and electromechanical transformer ratio, were analytically derived. By utilizing the array solution methods previously established for the thickness-mode piezoelectric devices and capacitive micromachined ultrasonic transducers (cMUTs), the single pMUT circuit model can be extended to models for array structures. The array model includes both the self- and mutual-acoustic radiation impedances of individual transducers in the acoustic medium. Volumetric displacement, induced piezoelectric current, and pressure field can be derived with respect to the input voltage matrix, material, and geometrical properties of each individual transducer and the array structure. As such, the analytical models presented here can be used as a guideline for analyses and design evaluations of large arrays of curved and flat pMUTs efficiently and can be further generalized to evaluate other pMUT architectures in the form of single devices or arrays.

Equivalent Circuit Models for Large Arrays of Curved and Flat Piezoelectric Micromachined Ultrasonic Transducers

Capacitive micromachined ultrasonic transducers (CMUTs) have great potential to compete with piezoelectric transducers in high-power applications. As the output pressures increase, nonlinearity of CMUT must be reconsidered and optimization is required to reduce harmonic distortions. In this paper, we describe a design approach in which uncollapsed CMUT array elements are sized so as to operate at the maximum radiation impedance and have gap heights such that the generated electrostatic force can sustain a plate displacement with full swing at the given drive amplitude. The pro- posed design enables high output pressures and low harmonic distortions at the output. An equivalent circuit model of the array is used that accurately simulates the uncollapsed mode of operation. The model facilities the design of CMUT parameters for high-pressure output, without the intensive need for computationally involved FEM tools. The optimized design requires a relatively thick plate compared with a conventional CMUT plate. Thus, we used a silicon wafer as the CMUT plate. The fabrication process involves an anodic bonding process for bonding the silicon plate with the glass substrate. To eliminate the bias voltage, which may cause charging problems, the CMUT array is driven with large continuous wave signals at half of the resonant frequency. The fabricated arrays are tested in an oil tank by applying a 125-V peak 5-cycle burst sinusoidal signal at 1.44 MHz. The applied voltage is in- creased until the plate is about to touch the bottom electrode to get the maximum peak displacement. The observed pressure is about 1.8 MPa with -28 dBc second harmonic at the surface of the array.

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High-power CMUTs: design and experimental verification

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.

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Electrostatic and small-signal analysis of CMUTs with circular and square anisotropic plates

This paper describes a correction and an extension in the previously published large signal equivalent circuit model for a circular capacitive micromachined ultrasonic transducer (CMUT) cell The force model is rederived so that the energy and power is preserved in the equivalent circuit model The model is able to predict the entire behavior of CMUT until the membrane touches the substrate Many intrinsic properties of the CMUT cell, such as the collapse condition, collapse voltage, the voltage–displacement interrelation and the force equilibrium before and after collapse voltage in the presence of external static force, are obtained as a direct consequence of the model The small signal equivalent circuit for any bias condition is obtained from the large signal model The model can be implemented in circuit simulation tools and model predictions are in excellent agreement with finite element method simulations

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An improved lumped element nonlinear circuit model for a circular CMUT cell

A very accurate equivalent circuit model for circular CMUT cell is recently proposed. As a consequence of this model, the operating characteristics of the cell is derived and presented as a biasing chart for CMUTs. We discuss these operating characteristics and show how the coupling coefficient can be derived from the model. We present an analysis of receiver performance of a CMUT cell. We also demonstrate how to design cells for fixed bias voltage but for different static depression.

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Designing circular CMUT cells using CMUT biasing chart

A nonlinear equivalent circuit for immersed transmitting capacitive micromachined ultrasonic transducers (CMUTs) is presented. The velocity profile across the CMUT surface maintains the same form over a wide frequency range. This property and the profile are used to model both the electromechanical conversion and the mechanical section. The model parameters are calculated considering the root mean square of the velocity distribution on the membrane surface as the through variable. The new model is compared with the FEM simulation results. The new model predicts the CMUT performance very accurately.

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A novel equivalent circuit model for CMUTs

An equivalent electrical circuit model valid for collapsed mode operation of CMUT is described. The across and through variables of the circuit model are chosen to be rms force and rms displacement over the surface of the CMUT membrane. The relation between rms displacement and applied voltage is obtained through analytical calculations utilizing the exact force distribution. The radiation impedance of collapsed mode CMUT is included as a load impedance in the circuit model. The resulting equivalent circuit is merged with uncollapsed mode model, to obtain a simulation tool that covers the whole operation range of CMUT.

/pdf/nonlinear-equivalent-circuit-model-for-circular-cmuts-in-4684h8q9cu.pdf

H. Kagan Oguz

Papers

An improved lumped element nonlinear circuit model for a circular CMUT cell

Equivalent circuit-based analysis of CMUT cell dynamics in arrays

Designing circular CMUT cells using CMUT biasing chart

A novel equivalent circuit model for CMUTs

Nonlinear equivalent circuit model for circular CMUTs in uncollapsed and collapsed mode