Electromechanical Equivalent Circuit Research Articles

Dynamic modeling and control of a conveyance microrobotic system using active friction drive

Presents a new generation of compliant multidegree of freedom piezoelectric microconveyer for microobjects based on the cooperation of arrayed direct-drive micro standing-wave ultrasonic actuators (microSWUMs). Their operating driving principles based on active frictional contact forces offer direct-drive and low-speed characteristics at the microscale. The tradeoff, however, is the complexity of dynamic modeling and control to cope with the optimization of the intermittent friction drive mechanism. A method using an equivalent electromechanical circuit is proposed for estimating and analyzing the optimum driving force, including the dynamic electrical and the mechanical energy conversions. On the basis of the proposed method, the friction drive optimization of the microrobot is performed through the implementation of different controllers: 1) an electromagnetic-field-based preload controller ensuring optimal preload; 2) an open-loop piecewise-modulated controller for self locking and driving force control; and 3) a resonant frequency compensation. Finally, an experimental investigation has been performed on a prototype of ultrasonic microconveyer incorporating 48 arrayed microSWUMs whose overall dimensions are 47 /spl times/ 29 mm/sup 2/ in order to demonstrate the proposed optimized friction drive.

Electromechanical coupling factor of capacitive micromachined ultrasonic transducers.

Recently, a linear, analytical distributed model for capacitive micromachined ultrasonic transducers (CMUTs) was presented, and an electromechanical equivalent circuit based on the theory reported was used to describe the behavior of the transducer [IEEE Trans. Ultrason. Ferroelectr. Freq. Control 49, 159-168 (2002)]. The distributed model is applied here to calculate the dynamic coupling factor k(w) of a lossless CMUT, based on a definition that involves the energies stored in a dynamic vibration cycle, and the results are compared with those obtained with a lumped model. A strong discrepancy is found between the two models as the bias voltage increases. The lumped model predicts an increasing dynamic k factor up to unity, whereas the distributed model predicts a more realistic saturation of this parameter to values substantially lower. It is demonstrated that the maximum value of k(w), corresponding to an operating point close to the diaphragm collapse, is 0.4 for a CMUT single cell with a circular membrane diaphragm and no parasitic capacitance (0.36 for a cell with a circular plate diaphragm). This means that the dynamic coupling factor of a CMUT is comparable to that of a piezoceramic plate oscillating in the thickness mode. Parasitic capacitance decreases the value of k(w), because it does not contribute to the energy conversion. The effective coupling factor k(eff) is also investigated, showing that this parameter coincides with k(w) within the lumped model approximation, but a quite different result is obtained if a computation is made with the more accurate distributed model. As a consequence, k(eff), which can be measured from the transducer electrical impedance, does not give a reliable value of the actual dynamic coupling factor.

Methods of experimental investigation of acoustic interaction between electroacoustical transducers in array

Determination of acoustic loading conditions on projectors in underwater arrays is needed to predict the performance of the array. Nonuniform distribution of the acoustic loading over the radiating surface of the array is generated due to acoustic interactions between the projectors, and often causes performance degradation of the directivity patterns and the transmit frequency response. In the electromechanical equivalent circuit model of such an array, the influence of the neighboring transducers can be considered as the mutual radiation impedance that is inserted in series with the self-radiation impedance. The mutual radiation impedance depends on the configuration of the transducer, the mode of vibration, velocity distribution, and the geometry of the array. We present experimental techniques to determine the mutual radiation impedance as an intrinsic property of a given transducer and array configuration. Two experimental methods are presented: the Z method and the V method. The methods are based on the measurements of the input impedances and on the output voltages of the interacting transducers. For the two methods, analytical investigations based on the electromechanical equivalent circuit analysis will be presented along with supporting experimental results on cylindrical transducers. [Work supported in part by ONR 321SS Lindberg and BTECH Acoustics.]

Quasistatic coupling coefficients for electrostrictive ceramics

A generalized definition of the coupling coefficient, useful for the evaluation of transducers that incorporate an electrostrictive active element, is introduced. The definition is expressed under quasistatic conditions and becomes zero when no bias is applied (assuming that the effects of remanence are negligible), and remains zero under zero bias even when a significant prestress is present. This reflects a property of the piezoelectric coupling coefficient, which vanishes when the ceramic becomes depoled. The behavior of this definition thus differs from that of another definition, introduced elsewhere, which produces a significant nonzero result at zero bias. [See C. Hom et al., “Calculation of quasi-static electromechanical coupling coefficients for electrostrictive ceramic materials,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 41, 542–551 (1994).] The present definition also leads in a natural way to a coupling coefficient for biased piezoelectric ceramics, and an equation is given for that case. Moreover, in the case of a biased electrostrictive ceramic it is found that a coupling coefficient derived from an equivalent circuit [J. C. Piquette and S. E. Forsythe, “Generalized material model for lead magnesium niobate (PMN) and an associated electromechanical equivalent circuit,” J. Acoust. Soc. Am. 104, 2763–2772 (1998)] gives an excellent approximation to the exact value, and is found to be accurate to within a few percent for drive amplitudes as high as 75% of the bias. It is shown that maximizing the coupling coefficient automatically discriminates against transducer designs (and operating conditions) that would produce significant harmonic distortion.

A defect coefficient — new insight into the responses of piezoelectric quartz crystal in gaseous and liquid phases

A defect coefficient ( d), defined as the ratio of an electromechanical coupling factor in liquid phase ( κ L 2) to that in gaseous phase ( κ g 2) has been proposed and applied to investigate the responses of a piezoelectric quartz crystal in gaseous and liquid phases. On the basis of an electromechanical equivalent circuit model, the value of d is calculated to be 1/2 and the explicit expression of κ L 2 is then given. According to electromechanical analogy, both sides of the expression L= m/ κ 2 are suggested to be divided into three terms when the crystal contacts with liquid. In this case, the corresponding expressions of κ 2 can also be obtained. By using κ L 2, R 1 given by Muramatsu is found to be 2 times as large as R 2 given by Martin, and this relationship is experimentally evaluated. Furthermore, the phenomenon that the crystal with a lower fundamental frequency ( f 0) gives larger response of R 1 than the crystal with a higher f 0 is rationally explained.

Thickness shearing vibration of the tangentially polarized piezoelectric ceramic thin circular ring

The thickness shearing vibration of the tangentially polarized piezoelectric ceramic thin circular ring is studied. The electrical and mechanical characteristics of the ring are analyzed in detail. The electro-mechanical equivalent circuit, the input electrical impedance, and the frequency equations for the resonance and anti-resonance frequencies are derived. A new concept that is characterized by the cross sectional shape and dimension is presented. An important conclusion is obtained that the thickness shearing vibration of the tangentially polarized piezoelectric ceramic ring is different from the traditional thickness extensional and radial extensional vibrations in that the electro-mechanical conversion coefficient, the electro-mechanical coupling coefficient, and the resonance frequency equation depend not only on the material parameters and the longitudinal dimension, but also on the lateral dimension, such as the cross sectional radius. It is shown that the measured resonance frequencies are in agreement with the theoretically calculated results, and the theoretical relationship between the resonance frequency and the cross sectional radius is also verified.

Experimental investigation of bending waves in ultrasonic reactors

A number of practical applications like ultrasonic cleaning, sonochemistry, and others use the effects of high, intense ultrasound. In general, the high-pressure sound fields can be obtained by different arrangements of piezoelectric compound transducers, which have to be optimized according to each application. A commonly used setup is a planar array transducer, consisting of one or more transducer elements. A tool to design and develop the transducer arrangements is given by simulations of the electromechanical equivalent circuit, although the results are often based on several assumptions. This paper reports on vibroacoustic investigations of compound transducer arrangements. Laser-optical measurements give evidence for arising bending waves on the radiating surface of the transducers that may affect the generated sound field and which have some effect on the total power distribution. The influence of bending waves has been included in the equivalent circuit; some simulations are presented.

Electromechanical Modelling Of Electrostatically Driven Micro Resonators, Part I: Basic Theory

Mechanical resonators operating in a flexural mode of vibration are examined theoretically with special regard to their electromechanical behaviour and equivalent electric circuits used to represent them. Electrostatic transducer elements are used for excitation and detection of the vibration. The differential equation of motion is solved in conjunction with the constitutive equations of the transducer elements. Arbitrary electrode shapes are accounted for in the model. The theory presented is exact with no restrictions other than linearity. General expressions of the network functions, e.g., the driving-point admittance of a one-port resonator, are derived, and subsequently used as a basis to develop equivalent electric circuits that accurately describe the resonator electromechanical behaviour. Resonators configured as an electric one-port as well as a two-port are considered in detail. Extensions of the equivalent circuit representation to accomodate N-ports are indicated.

A multi-mode piezomotor using a flextensional coupler

A new structure of piezomotor using a flextensional coupler is proposed. Two longitudinal actuators and an elliptical shell constitute its frame. The basic structure and the operating mode are first presented. The coupling between a translation mode and a flexion mode generates an elliptic motion at shell tops. The design method based on a finite element model and an electromechanical equivalent circuit permits the structure dimensioning. Construction of motor prototypes indicates the reliability of the principle, a good agreement with design results and interesting motor performances.

Generalized material model for lead magnesium niobate (PMN) and an associated electromechanical equivalent circuit

An existing one-dimensional nonlinear material model of lead magnesium niobate [J. C. Piquette and S. E. Forsythe, “A nonlinear model of lead magnesium niobate (PMN),” J Acoust. Soc. Am. 101, 289–296 (1997)] is generalized to three dimensions. The resulting theory is applied to two practical systems: the “thickness expander plate” and the “length expander bar.” Linearizing the theory results in an electromechanical equivalent circuit that is applicable to predicting the first-order behavior of transducers based on either of these practical systems. The methods used are sufficiently general that the circuit is also appropriate for piezoelectric, and even for electrostatic, transducers. Preliminary experimental data that confirm the validity of the circuit are presented. Connections between the constants of the theory and those of piezoelectricity are derived, and a general expression for the coupling coefficient is obtained. Known theoretical coupling coefficients for piezoelectric and electrostatic transducers are recovered as special cases. The coupling coefficient of the PMN bar at zero prestress and zero remanent polarization is examined in some detail. For this case it is found that, to a good approximation, the bias voltage of the optimal (i.e., maximum) coupling coefficient is √ times the bias voltage which produces the maximum effective piezoelectric d33. A simple formula for estimating second harmonic distortion at zero stress and remanent polarization is given. The D field for the bar is obtained in terms of a Maclaurin series in stress. This series is asymptotic in the electric field, and is useful for estimating harmonic distortion when the stress or remanent polarization are nonzero.

Analysis and Simulation of Stacked-Segment Electromechanical Transducers with Partial Electrical Excitation by PSPICE

A composite transducer is fabricated, by stacking certain segments of poled piezoelectric ceramic, to satisfy multiple, conflicting design specifications. An electromechanical equivalent circuit that substantially facilitates the analysis of segmented systems, based on the Mason model is derived. To obtain the optimal solution to a specific design formulation, PSPICE programming techniques can be applied. An example of the application of PSPICE programming in the segmented transducer analysis is presented and the simulation results are in good agreement with the experimental measurements.

Principles of operation of condenser microphones

Following a description of microphone cartridge construction, the topics to be discussed include: electrical and mechanical microphone sensitivities, effect of the polarization voltage, electrical and mechanical stability, membrane damping, electromechanical equivalent circuit, background noise, and response to infrasound. The discussion will be interspersed with little-known factoids on microphone performance which are not found in the traditional literature.

Torsional vibration of coaxially segmented, tangentially polarized piezoelectric ceramic tubes

Piezoelectric ceramic materials are often used in ultrasonic and underwater sound transducers by stacking a number of identical pieces together. In this paper, the theoretical behavior is derived for a piezoelectric ceramic cylindrical tube consisting of any number of tangentially polarized coaxial segments. The resonance frequency equation of the torsional tube is obtained and the electromechanical equivalent circuits are developed that greatly simplify and improve the analysis of composite resonators in transducer design. A new parameter which is defined as the cross-sectional torsional coefficient is presented. The relations between the theoretical parameters and the experimentally measured quantities of the segmented electromechanical system in torsional vibration are obtained so that the material parameters can be evaluated even though the lateral dimensions are large relative to the length of each individual segment and the analysis of the vibrational characteristics of the torsional transducer can be simplified.

Study of the sandwiched piezoelectric ultrasonic torsional transducer

The sandwiched piezoelectric ultrasonic torsional transducer was studied. The transducer consists of front and back metal cylinders, and coaxially segmented, tangentially polarized, piezoelectric ceramic tubes. The torsional vibration of the tangentially polarized piezoelectric ceramic slender tube was studied first and its electromechanical equivalent circuit was derived. Based on the network theory and the electromechanical equivalent circuit, the torsional vibration of the piezoelectric ceramic cylinder, formed by stacking a number of identical short piezoelectric ceramic rings, was analysed and the electromechanical equivalent circuit of the piezoelectric ceramic stack in torsional vibration was developed. Finally, the sandwiched ultrasonic torsional transducer was studied and resonance frequency equations were derived which can be used to design and calculate the torsional transducers for different applications.

Numerical models used in design and analysis of volumetric transducer arrays

Analysis of transducer element and array performance has been aided by development of computer-based numerical models that include computation of mutual coupling effects. The farfield pressure distribution and array source level calculations necessitate accurate modeling of the individual transducers (in free field) as well as the varying radiation impedances due to the acoustic coupling between elements. Electromechanical equivalent circuits are employed to determine the complex drive voltages required to achieve velocity control within the array. Optimization methods that result in increased array source level and efficiency without forsaking velocity control have been investigated. This paper discusses numerical computations of farfield pressure, acoustic power, and mutual radiation impedances within volumetric projector arrays. Methods of optimization are also discussed. Predicted characteristics such as beam patterns and source level curves for various arrays of low-frequency flexural disk and flextensional transducers will be presented and compared to measurements.

Optimization of a low-frequency transmitting array

A low-frequency plane array is studied in water in a wide frequency band around the resonance frequency of the transducers. The array under study is made of eight length-expander vibrators (two columns of four transducers) with a circular radiating face in a rigid box of limited dimensions. The radiating impedance matrices are calculated by an integral equation method [C. Audoly, J. Acoust. Soc. Am. Suppl. 1 83, S20 (1988)] and projectors are modeled with a classical electromechanical equivalent circuit. Due to the effects of acoustic interactions, no terms in the matrices are found to be negligible. Mechanical and electrical constraints on the transducers are identified and computed. The array is studied under three conditions: identical voltage driving, identical headmass velocity distribution, and acoustic power optimization. The results confirm that acoustic interactions have important and drastic effects around the resonance. The study of acoustic power optimization makes it possible to discuss the opportunity of using velocity control and electromechanical feedback devices in low-frequency sonar projector arrays.

Optimization of a low-frequency transmitting array

A low-frequency plane array is studied in water in a wide frequency band around the resonance frequency of the transducers. The array under study is made of eight length-expander vibrators (two columns of four transducers) with a circular radiating face in a rigid box of limited dimensions. The radiating impedance matrices are calculated by an integral equation method [C. Audoly, J. Acoust. Soc. Am. Suppl. 1 83, S20 (1988)] and projectors are modeled with a classical electromechanical equivalent circuit. Due to the effects of acoustic interactions, no terms in the matrices are found to be negligible. Mechanical and electrical constraints on the transducers are identified and computed. The array is studied under three conditions: identical voltage driving, identical headmass velocity distribution, and acoustic power optimization. The results confirm that acoustic interactions have important and drastic effects around the resonance. The study of acoustic power optimization makes it possible to discuss the opportunity of using velocity control and electromechanical feedback devices in low-frequency sonar projector arrays.

Tunable sandwich transducer

A tunable sandwich transducer is reported which has its fundamental resonant frequency controlled by varying a passive reactive load. The structure consists of four PZT-4 rings sandwiched between an aluminium head-mass and a brass tail-mass. The electromechanical coupling coefficient and the radiation efficiency into water of the transducer are given over the frequency range 25 to 75 kHz. A theoretical analysis based on an electromechanical equivalent circuit shows good agreement with the experimental results. >

Analysis of multilayered wideband transducers

Single piezoelectric material used as a transducer element usually gives a high Q resonance characteristic due to high acoustic impedance relative to that of water. Epoxy resins or similar materials are used as impedance matching layers in order to obtain a low Q transducer. Although analysis of a composite structure element is traditionally made with an electromechanical equivalent circuit, numerical analysis is increasingly in use. In this paper, a simple and useful design method for a multilayered transducer utilizing both an equivalent circuit and a finite element method is described. Transducers with a half‐wavelength piezoelectric material with one or two quarter wavelength matching layers are analyzed for transient response with pulsed sine wave input as well as for steady‐state response, and vibration, stress, and sound radiation characteristics are obtained. The results indicate that the best matching impedance is different from the value usually suggested [B. V. Smith and B. K. Gazey, IEE Proc. 113, 285–297 (1984)].

Equivalent Mechanical Circuits of Tuning Fork Angular Rate Gyros

Approximate equivalent mechanical circuits for some types of tuning fork gyros, that is, Sperry type, Watson type and a double resonant tuning fork type are determined and compared. A resultant equivalent electro-mechanical circuit of piezoelectric vibratory gyros is also shown, as is an equivalent mechanical circuit of a rotor gyro.