Unbiased Charged Circular CMUT Microphone: Lumped-Element Modeling and Performance.

An energy-consistent lumped-element equivalent circuit model for charged circular capacitive micromachined ultrasonic transducer (CMUT) cell is derived and presented. It is analytically shown and experimentally verified that a series dc voltage source at the electrical terminals is sufficient to model the charging in CMUT. A model-based method for determining this potential from impedance measurements at low bias voltages is presented. The model is validated experimentally using an airborne CMUT, which resonates at 103 kHz. Impedance measurements, reception measurements at resonance and off-resonance, and the transient response of the CMUT are compared with the model predictions.

Bilateral CMUT Cells and Arrays: Equivalent Circuits, Diffraction Constants, and Substrate Impedance.

We introduce the large-signal and small-signal equivalent circuit models for a capacitive micromachined ultrasonic transducer (CMUT) cell, which has radiating plates on both sides. We present the diffraction coefficient of baffled and unbaffled CMUT cells. We show that the substrate can be modeled as a very thick radiating plate on one side, which can be readily incorporated in the introduced model. In the limiting case, the reactance of this backing impedance is entirely compliant for substrate materials with a Poisson's ratio less than 1/3. We assess the dependence of the radiation performance of the front plate on the thickness of the back plate by simulating an array of bilateral CMUT cells. We find that the small-signal linear model is sufficiently accurate for large-signal excitation, for the purpose of the determining the fundamental component. To determine harmonic distortion, the large-signal model must be used with harmonic balance analysis. Rayleigh-Bloch waves are excited at the front and back surfaces similar to conventional CMUT arrays.

Rayleigh-Bloch waves in CMUT arrays.

Using the small-signal electrical equivalent circuit of a capacitive micromachined ultrasonic transducer (CMUT) cell, along with the self and mutual radiation impedances of such cells, we present a computationally efficient method to predict the frequency response of a large CMUT element or array. The simulations show spurious resonances, which may degrade the performance of the array. We show that these unwanted resonances are due to dispersive Rayleigh-Bloch waves excited on the CMUT surface-liquid interface. We derive the dispersion relation of these waves for the purpose of predicting the resonance frequencies. The waves form standing waves at frequencies where the reflections from the edges of the element or the array result in a Fabry-Pérot resonator. High-order resonances are eliminated by a small loss in the individual cells, but low-order resonances remain even in the presence of significant loss. These resonances are reduced to tolerable levels when CMUT cells are built from larger and thicker plates at the expense of reduced bandwidth.

Designing transmitting CMUT cells for airborne applications.

We report a new mode of airborne operation for capacitive micromachined ultrasonic transducers (CMUT), in which the plate motion spans the entire gap without collapsing and the transducer is driven by a sinusoidal voltage without a dc bias. We present equivalent-circuit-based design fundamentals for an airborne CMUT cell and verify the design targets using fabricated CMUTs. The performance limits for silicon plates are derived. We experimentally obtain 78.9 dB//20 µPa@1 m source level at 73.7 kHz, with a CMUT cell of radius 2.05 mm driven by 71 V sinusoidal drive voltage at half the frequency. The measured quality factor is 120. We also study and discuss the interaction of the nonlinear transduction force and the nonlinearity of the plate compliance.

Parametric nonlinear lumped element model for circular CMUTs in collapsed mode.

We present a parametric equivalent circuit model for a circular CMUT in collapsed mode. First, we calculate the collapsed membrane deflection, utilizing the exact electrical force distribution in the analytical formulation of membrane deflection. Then we develop a lumped element model of collapsed membrane operation. The radiation impedance for collapsed mode is also included in the model. The model is merged with the uncollapsed mode model to obtain a simulation tool that handles all CMUT behavior, in transmit or receive. Large- and small-signal operation of a single CMUT can be fully simulated for any excitation regime. The results are in good agreement with FEM simulations.

Equivalent circuit-based analysis of CMUT cell dynamics in arrays.

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.

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

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.

High-power CMUTs: design and experimental verification.

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 proposed 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 increased 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.

Radiation impedance of collapsed capacitive micromachined ultrasonic transducers.

The radiation impedance of a capacitive micromachined ultrasonic transducer (CMUT) array is a critical parameter to achieve high performance. In this paper, we present a calculation of the radiation impedance of collapsed, clamped, circular CMUTs both analytically and using finite element method (FEM) simulations. First, we model the radiation impedance of a single collapsed CMUT cell analytically by expressing its velocity profile as a linear combination of special functions for which the generated pressures are known. For an array of collapsed CMUT cells, the mutual impedance between the cells is also taken into account. The radiation impedances for arrays of 7, 19, 37, and 61 circular collapsed CMUT cells for different contact radii are calculated both analytically and by FEM simulations. The radiation resistance of an array reaches a plateau and maintains this level for a wide frequency range. The variation of radiation reactance with respect to frequency indicates an inductance-like behavior in the same frequency range. We find that the peak radiation resistance value is reached at higher kd values in the collapsed case as compared with the uncollapsed case, where k is the wavenumber and d is the center-to-center distance between two neighboring CMUT cells.

Deep-collapse operation of capacitive micromachined ultrasonic transducers.

Capacitive micromachined ultrasonic transducers (CMUTs) have been introduced as a promising technology for ultrasound imaging and therapeutic ultrasound applications which require high transmitted pressures for increased penetration, high signal-to-noise ratio, and fast heating. However, output power limitation of CMUTs compared with piezoelectrics has been a major drawback. In this work, we show that the output pressure of CMUTs can be significantly increased by deep-collapse operation, which utilizes an electrical pulse excitation much higher than the collapse voltage. We extend the analyses made for CMUTs working in the conventional (uncollapsed) region to the collapsed region and experimentally verify the findings. The static deflection profile of a collapsed membrane is calculated by an analytical approach within 0.6% error when compared with static, electromechanical finite element method (FEM) simulations. The electrical and mechanical restoring forces acting on a collapsed membrane are calculated. It is demonstrated that the stored mechanical energy and the electrical energy increase nonlinearly with increasing pulse amplitude if the membrane has a full-coverage top electrode. Utilizing higher restoring and electrical forces in the deep-collapsed region, we measure 3.5 MPa peak-to-peak pressure centered at 6.8 MHz with a 106% fractional bandwidth at the surface of the transducer with a collapse voltage of 35 V, when the pulse amplitude is 160 V. The experimental results are verified using transient FEM simulations.

An equivalent circuit model for transmitting capacitive micromachined ultrasonic transducers in collapse mode.

The collapse mode of operation of capacitive micromachined ultrasonic transducers (CMUTs) was shown to be a very effective way to achieve high output pressures. However, no accurate analytical or equivalent circuit model exists for understanding the mechanics and limits of the collapse mode. In this work, we develop an equivalent nonlinear electrical circuit that can accurately simulate the mechanical behavior of a CMUT with given dimensions and mechanical parameters under any large or small signal electrical excitation, including the collapse mode. The static and dynamic deflections of a plate predicted from the model are compared with finite element simulations. The equivalent circuit model can estimate the static deflection and transient behavior of a CMUT plate to within 5% accuracy. The circuit model is in good agreement with experimental results of pulse excitation applied to fabricated CMUTs. The model is suitable as a powerful design and optimization tool for collapsed and uncollapsed CMUTs.

Radiation impedance of an array of circular capacitive micromachined ultrasonic transducers.

The radiation impedance of a capacitive micromachined ultrasonic transducer (cMUT) with a circular membrane is calculated analytically using its velocity profile for the frequencies up to its parallel resonance frequency for both the immersion and the airborne applications. The results are verified by finite element simulations. The work is extended to calculate the radiation impedance of an array of cMUT cells positioned in a hexagonal pattern. A higher radiation resistance improves the bandwidth as well as the efficiency of the cMUT. The radiation resistance is determined to be a strong function of the cell spacing. It is shown that a center-to-center cell spacing of 1.25 wavelengths maximizes the radiation resistance, if the membranes are not too thin. It is also found that excitation of nonsymmetric modes may reduce the radiation resistance in immersion applications.

Nonlinear modeling of an immersed transmitting capacitive micromachined ultrasonic transducer for harmonic balance analysis.

Finite element method (FEM) is used for transient dynamic analysis of capacitive micromachined ultrasonic transducers (CMUT) and is particularly useful when the membranes are driven in the nonlinear regime. One major disadvantage of FEM is the excessive time required for simulation. Harmonic balance (HB) analysis, on the other hand, provides an accurate estimate of the steady-state response of nonlinear circuits very quickly. It is common to use Mason's equivalent circuit to model the mechanical section of CMUT. However, it is not appropriate to terminate Mason's mechanical LC section by a rigid piston's radiation impedance, especially for an immersed CMUT. We studied the membrane behavior using a transient FEM analysis and found out that for a wide range of harmonics around the series resonance, the membrane displacement can be modeled as a clamped radiator. We considered the root mean square of the velocity distribution on the membrane surface as the circuit variable rather than the average velocity. With this definition, the kinetic energy of the membrane mass is the same as that in the model. We derived the force and current equations for a clamped radiator and implemented them using a commercial HB simulator. We observed much better agreement between FEM and the proposed equivalent model, compared with the conventional model.

Reducing anchor loss in micromechanical extensional mode resonators.

In this work, we propose a novel method to increase the quality factor of extensional mode micromechanical resonators. The proposed resonator topology is suitable for integration in a silicon-based process to fabricate micromechanical filters and oscillators. It is a half-wavelength-long strip excited longitudinally by electrostatic forces, and it is isolated from the substrate by alternating with bars of a quarter wavelength long. This structure causes a large impedance mismatch between the resonator and the substrate and hence reduces the anchor loss considerably. The performance of the resonator is determined by finite element simulations. We introduce an equivalent electrical circuit to predict the performance of the resonator. The electrical model gives results consistent with the finite element simulations. The proposed resonator is expected to have a very small anchor loss resulting in a very high Q.

Tunable surface plasmon resonance on an elastomeric substrate.

In this study, we demonstrate that periods of metallic gratings on elastomeric substrates can be tuned with external strain and hence are found to control the resonance condition of surface plasmon polaritons. We have excited the plasmon resonance on the elastomeric grating coated with gold and silver. The grating period is increased up to 25% by applying an external mechanical strain. The tunability of the elastomeric substrate provides the opportunity to use such gratings as efficient surface enhanced Raman spectroscopy substrates. It's been demonstrated that the Raman signal can be maximized by applying an external mechanical strain to the elastomeric grating.

Parametric linear modeling of circular cMUT membranes in vacuum.

We present a lumped element parametric model for the clamped circular membrane of a capacitive micromachined ultrasonic transducer (cMUT). The model incorporates an electrical port and two sets of acoustic ports, through which the cMUT couples to the medium. The modeling approach is based on matching a lumped element model and the mechanical impedance of the cMUT membrane at the resonance frequencies in vacuum. Very good agreement between finite element simulation results and model impedance is obtained. Equivalent circuit model parameters can be found from material properties and membrane dimensions without a need for finite element simulation.

Optimization of the gain-bandwidth product of capacitive micromachined ultrasonic transducers.

Capacitive micromachined ultrasonic transducers (cMUT) have large bandwidths, but they typically have low conversion efficiencies. This paper defines a performance measure in the form of a gain-bandwidth product and investigates the conditions in which this performance measure is maximized. A Mason model corrected with finite-element simulations is used for the purpose of optimizing parameters. There are different performance measures for transducers operating in transmit, receive, or pulse-echo modes. Basic parameters of the transducer are optimized for those operating modes. Optimized values for a cMUT with silicon nitride membrane and immersed in water are given. The effect of including an electrical matching network is considered. In particular, the effect of a shunt inductor in the gain-bandwidth product is investigated. Design tools are introduced, which are used to determine optimal dimensions of cMUTs with the specified frequency or gain response.