A history of equivalent circuit modeling of acoustic structures is presented. This is followed by a tutorial development of their use to represent piezoelectric (PE) and piezomagnetic (PM) plate transducers and bimorph cantilever beams for the purpose of facilitating transition of modern micro/nanotechnology creations to practical sensor, actuator, and transducer applications. Circuit approximations of various types are derived from the more general networks.

Modeling piezoelectric and piezomagnetic devices and structures via equivalent networks

A tutorial synopsis of the piezoelectric effect is presented in the context of its history, traditional uses, and relation to crystal symmetry. Associated effects are briefly noted. Future prospects, particularly in the area of microelectromechanical systems/structures (MEMS) are discussed. >

Piezoelectricity: old effect, new thrusts

This paper focuses on the design, fabrication and characterization of unimorph actuators for a microaerial flapping mechanism. PZT-SH and PZN-PT are investigated as piezoelectric layers in the unimorph actuators. Design issues for microaerial flapping actuators are discussed, and criteria for the optimal dimensions of actuators are determined. For low power consumption actuation, a square wave based electronic driving circuit is proposed. Fabricated piezoelectric unimorphs are characterized by an optical measurement system in quasi-static and dynamic mode. Experimental performance of PZT-5H and PZIV-PT based unimorphs is compared with desired design specifications. A 1-DOF flapping mechanism with a PZT-SH unimorph is constructed, and 180/spl deg/ stroke motion at 95 Hz is achieved. Thus, it is shown that unimorphs could be promising flapping mechanism actuators.

Development of PZT and PZN-PT based unimorph actuators for micromechanical flapping mechanisms

The aim of this paper is the modelling of a non-symmetric bimorph constituted by a piezoelectric material deposited on an alumina substrate and used either as an actuator or a sensor. Theoretical modelling based on the flexural modes of the structure is carried out and the influence of the electrode characteristics (geometrical dimensions and elastic parameters) is introduced in the modelling for calculating the bimorph bending displacement. In actuator mode, the electrical admittance of the cantilever non-symmetric bimorph is stated and the intrinsic electromechanical coupling factor linked to the bimorph bending motion is deduced and compared with that defined in IEEE Standards. The analytical modelling was used for characterizing a cantilever bimorph constituted by a piezoelectric thick film deposited on an alumina substrate. A trial and error fitting method is described for determining the elastic, piezoelectric and dielectric constants of the piezoelectric material. The influence of the electrode parameters is calculated and the measurement uncertainty is deduced. In sensor mode the open voltage delivered by the bent piezoelectric layer and the electrical equivalent circuit of the bimorph are given. Theoretical results are compared with those obtained by the finite element method, and discussed.

https://iopscience.iop.org/article/10.1088/0960-1317/13/6/306/pdf

Modelling of a cantilever non-symmetric piezoelectric bimorph

New thick film ceramic processing techniques and microelectromechanical systems require material characterization of a piezoelectric film supported by a substrate. An analytical solution of the one-dimensional wave equation for multiple layered systems driven in the thickness mode is presented. The impedance across the piezoelectric layer is derived and expressed in terms of the material properties of the two materials. This includes the open-circuit elastic stiffness c33D, the clamped permittivity e33S and the h33 piezoelectric constant of the piezoelectric layer and the elastic stiffness csD of the substrate. The properties are expressed as complex variables in order to account for the losses within the materials. The material parameters of the solution are extracted from experimental results using a modified Levenberg–Marquardt technique. The capabilities of this nondestructive technique are demonstrated using experimental and simulated impedance spectra of lead zirconium titanate sol gel composite coa...

Thickness mode material constants of a supported piezoelectric film

The resonances of dynamically excited symmetric piezoelectric bimorphs have been determined from the equations of state Under the effect of sinusoidal stimuli: a moment exerted at the tip M, a force exerted perpendicular to the plane of the bimorph also applied at the tip F, a uniformly applied pressure p, and an electrode voltage V, they respond with a sinusoidal tip rotation /spl alpha/, tip deflection /spl delta/, volume displacement /spl nu/, and electrode charge Q All of the former are related to all of the latter through a dynamic admittance matrix B The antiresonance frequency of the capacitance C have been found while also antiresonance in off-diagonal elements have been determined The latter indicate that at these frequencies the bimorph does not work as an actuator or sensor in the particular domain of the off-diagonal The mode shape at these antiresonance frequencies has been determined The antiresonance of b/sub 14/ determines that for this frequency the tip has deflection but no rotation, while the antiresonance of b/sub 24/ indicates that the tip has rotation but no deflection No antiresonance in the volume displacement is found, indicating that the bimorph is a pressure converter (microphone) at all frequencies Micromachined piezoelectric heterogeneous bimorphs have been fabricated using the techniques of IC fabrication Their deflections have been measured as a function of frequency and applied voltage, while these have been compared with the theoretical predictions An anomalously large quadratic deflection has been found, superimposed on the linear piezoelectric behavior The agreement between the linear part of the experimental deflection and the theory was quite good

Resonance and antiresonance of symmetric and asymmetric cantilevered piezoelectric flexors

The constituent equations for piezoelectric bimorphs are written in terms of redefined extensive and intensive variables which makes it possible to write the bimorph matrix in numerical form. This numerical matrix is diagonalized to give the eigenvalues of bimorphs. The modal matrix is used to find the corresponding eigenstates. A physical interpretation of bimorph eigenstates is proposed. Two realizations of network representation are given for the admittance matrix. The eigenvalue matrix is inverted to give the impedance matrix. The network representation of the impedance matrix is given. >

Simulation networks for piezoelectric bimorph actuators

The elastic modulus of thin film ZnO has been measured by attaching a Si-ZnO strip to a ceramic bimorph, deflecting the latter with an electric signal, and measuring the resonance frequencies of the Si-ZnO strip. The resonance frequencies of the Si-ZnO strip depend on the properties of both Si and ZnO. The properties of Si are known, and, thus, the elastic modulus of thin film ZnO can be calculated from the resonance frequency of the beam. The results we obtained are presented in this report.

Measurement of elastic modulus s/sub 11//sup D/ of thin film ZnO by resonance method

Micromachined piezoelectric heterogeneous bimorphs have been fabricated using the techniques of IC fabrication. The voltage across the piezoelectric element of a heterogeneous bender causes it to contract or expand while the passive element is compressed or elongated and responds with a counterforce, thus building up a moment, which bends the structure. This dynamic moment has been calculated, and the corresponding Euler-Bernoulli equation has been written with a time-dependent voltage as a source term in the inhomogeneous part of the equation thus becoming a Rayleigh equation. The bending resistance has been determined using the transformation of area method. Once the solution of this equation is found the deformation can be calculated and the resulting dielectric displacement can be determined. From the latter we can determine the charge that flows into the electrodes, by integrating over the surface area. The relation between the charge on the electrodes and the driving voltage is the capacitance C. The capacitance has the expected resonance behavior: the bimorph goes through resonance at frequencies which are the solutions of the implicit relation cos/spl Omega/L cosh/spl Omega/L=-1. The deflection of the bimorph has also been calculated and follows a similar behavior. >

J.G. Smits

Papers

Resonance and antiresonance of symmetric and asymmetric cantilevered piezoelectric flexors

Simulation networks for piezoelectric bimorph actuators

Measurement of elastic modulus s/sub 11//sup D/ of thin film ZnO by resonance method

Dynamic deflection and capacitance of micromachined piezoelectric benders