Electromechanical Equivalent Circuit Research Articles

Theoretical modeling of triboelectric receiver transducer for mechanic-electrical transformations

In recent years, receiver transducers have gained extensive attention in both research and industrial fields. In particular, the triboelectric receiver transducer (TE-RT) stands out due to their self-powered capability, low cost, and excellent output performance. While theoretical methods for analyzing the output performance are still lacked, which may limit the further development of this field. In this study, we propose an equivalent electromechanical conversion circuit that couples the acoustic field and electric field to predict the performance characteristics of the TE-RT. Specifically, the relationship between the deformation of the diaphragm and output performance of the TE-RT is analyzed through theoretical modeling and clarification of governing equations. Subsequently, a specific transformer revealing the mechanical-electrical conversion mechanism for the TE-RT was proposed using the equivalent circuit method (ECM), and an ECM model was established to analyze the performance of TE-RT quickly and accurately. To further improve the output performance of the TE-RT, piezoelectric materials are integrated into the model to enhance its capacity for converting acoustic energy. Verification for the two types of receiver transducer was conducted using the finite element method (FEM), and the equivalent circuit model fit the simulation results well. The acoustic energy conversion efficiency of the TE-RT can also be further evaluated by the number of turns (n) in the specialized transformer. This study can provide a solid theoretical foundation for the design and development of receiver transducer.

Measurement insights and error analysis of electronic parameters for ultrasonic transducers

Abstract Piezoelectric ultrasonic transducers with the function of energy conversation, as well as their numerous advantages in high-power density, quick response, flexible design, and service reliability, are involved in wide applications of industrial processing, precision driving, smart sensing, and medical services. The electromechanical equivalent circuit and Kirchhoff’s law indicate that mechatronics parameters are essential for performance evaluation, reliability analysis, and fault diagnosis of ultrasonic transducers. Importantly, the ultrasonic transducer is a time-variant system, data of one single parameter collected from a certain test cannot match with the data of another single parameter acquired from a different test. So, a synchronous and precise online measurement of electronic parameters is encouraged for performance evaluation and optimization design of ultrasonic transducers. With the combination of virtual instrument technology, an asynchronous measurement system of electrical excitation parameters for the ultrasonic transducers of linear driving motors was established in this study. Furthermore, the systematic measurement methods and error theory were illustrated, including the calculation methods and measuring circuits of electric signals. Experimental results proved that the proposed system and methods for measuring the input electronic power of ultrasonic transducer (e.g. effective value method for voltage and current, energy moment method for frequency, and spectrum analysis method for phase difference) are highly precise, quickly responsive, robust, and reliable for ultrasonic transducers. The findings of this study provide valuable references and suggestions for efficient, accurate, and online performance evaluation of ultrasonic transducers, particularly for piezoelectric transducers utilizing ultrasonic high-voltage exciting signals.

Electro-Mechanical Characterization and Modeling of a Broadband Piezoelectric Microgenerator Based on Lithium Niobate.

Vibration energy harvesting based on piezoelectric transducers is an attractive choice to replace single-use batteries in powering Wireless Sensor Nodes (WSNs). As of today, their widespread application is hindered due to low operational bandwidth and the conventional use of lead-based materials. The Restriction of Hazardous Substances legislation (RoHS) implemented in the European Union restricts the use of lead-based piezoelectric materials in future electronic devices. This paper investigates lithium niobate (LiNbO3) as a lead-free material for a high-performance broadband Piezoelectric Energy Harvester (PEH). A single-clamped, cantilever beam-based piezoelectric microgenerator with a mechanical footprint of 1 cm2, working at a low resonant frequency of 200 Hz, with a high piezoelectric coupling coefficient and broad bandwidth, was designed and microfabricated, and its performance was evaluated. The PEH device, with an acceleration of 1 g delivers a maximum output RMS power of nearly 35 μW/cm2 and a peak voltage of 6 V for an optimal load resistance at resonance. Thanks to a high squared piezoelectric electro-mechanical coupling coefficient (k2), the device offers a broadband operating frequency range above 10% of the central frequency. The Mason electro-mechanical equivalent circuit was derived, and a SPICE model of the device was compared with experimental results. Finally, the output voltage of the harvester was rectified to provide a DC output stored on a capacitor, and it was regulated and used to power an IoT node at an acceleration of as low as 0.5 g.

A Timoshenko-Ehrenfest beam model for simulating Langevin transducer dynamics

The Langevin transducer is a widely used power ultrasonic device across both medical and industrial applications, from orthopaedic surgery to drilling and welding. It is a sandwich-type device which typically consists of piezoelectric ceramic rings between two metallic end-masses. The transducer is commonly operated at a particular resonant mode to deliver ultrasonic vibrations to a target structure of interest, and is generally modelled using approaches including the transfer matrix method and electromechanical equivalent circuit methods. Here, we propose a variational framework to study the dynamics of the Langevin transducer based on the Timoshenko-Ehrenfest beam theory and Hamilton's principle. The variational equation derived from this model is then discretized by the standard finite element method with spectral elements. To verify the proposed one-dimensional electromechanical model, the computed resonance frequencies, or natural frequencies, of the one-dimensional model are compared to those of a three-dimensional finite element model with respect to varying geometry parameters characterizing the transducer. The results of the reduced one-dimensional and full three-dimensional models are then compared to those measured through an experimental investigation consisting of laser Doppler vibrometry. This is undertaken for the first ten eigenfrequencies, including longitudinal and bending modes, where normalized amplitudes and vibration mode shapes are reported. The close correlation between modelling and experiment demonstrates that the proposed one-dimensional electromechanical model can deliver results consistent with the full three-dimensional model and from experiment, thus verifying a rapid and reliable method for studying the free vibrations and dynamics of the Langevin transducer which accounts for the axial vibrations of the piezoelectric ceramic stack in all cases.

Multidomain modeling of acoustical transducers and arrays using electrical network theory

Modeling of piezoceramic acoustic transducers involves the transformation of energy in the electrical, mechanical, acoustical radiation domains. By using generalized coordinates and Lagrangian mechanics, in which principle of least action is applied in each domain, the electrical, piezoelectric, mechanical and sound radiation problems can be solved separately and then combined in a multi-contour equivalent electro-mechanical circuit with each mechanical vibrational resonant modes representing separate degrees of freedom in the coupled electrical circuit. This energy approach involves the calculation of the potential and kinetic energies of each practical mode of vibration, which can be determined analytically, experimentally, or by finite-element-analysis. This is an alternative to the use of Newtonian mechanics which requires an accurate description of the real boundary conditions in the device and is common in many FEA modeling approaches. The availability of software (e.g., Matlab, Python, LTSpice, etc.) to model electrical circuits (networks in the case of multi-resonant devices) makes for a powerful and efficient modeling approach. The historical emergence of this approach stems from adapting the Rayleigh-Ritz method for mechanical and electrical domains with the advent of piezoelectric and magnetostrictive bodies. Several example of piezoceramic transducers are presented.

Radial vibration analysis for piezoceramic shell-stacked spherical transducer with thick walls

With the development of ultrasonic transducers, spherical piezoelectric transducers have attracted tremendous attention in a variety of application fields due to their ability to resist higher pressures and produce omnidirectional radiation. However, the wall thickness of piezoceramic spherical shells is usually thin due to the limitations of polarization technology and operating voltage, leading to the limited vibration performance and power capacity of the spherical transducer. We present a piezoceramic shell-stacked spherical transducer (PSST) capable of addressing the problem of difficult excitation caused by the thick wall of the piezoceramic shell. The resulting device consists of a two-layered piezoceramic shell interposed between the inner and outer concentric spherical metal shells. By removing the equivalent mechanical transformers, a novel electromechanical equivalent circuit of the PSST is established to simplify the theoretical analysis of the designed PSST. The electromechanical characteristics of the resulting device are experimentally verified, and the measured results are in good agreement with the theoretical predictions and simulation results. Our design opens up possibilities for designing spherical transducers with high-vibration performance and may offer potential for a wide range of applications such as underwater detection and structural health monitoring.

Ultrasonic tool shank with multiple vibration mode for micro-nano drilling: Design, optimization and experiment

The machining effects of hole drilling is limited by merely changing the amplitude. Adjusting ultrasonic vibration frequency to expand the matching range of machining parameters optimization is necessary to improve hole drilling effects. Therefore, the design, optimization, and experiment evaluation for a novel integrated, fined, low-cost dual-frequency ultrasonic tool shank is proposed, which can work in three vibration modes for micro-nano precision drilling. The principle of the dual-frequency ultrasonic transducer is introduced. The electro-mechanical equivalent circuit model with step horn is applied to study the dual-frequency characteristics. The transducer displacement equation model based on the wave equation theory is deduced to find the shared vibration node of the first and third frequency. Sensitivity analysis by equivalent circuit model and Finite element method (FEM) simulation of the key parameter is both conducted to optimize ultrasonic transducer geometry dimension. The vibration node of mounting flange is specified in an identical position for the dual vibration modes. A prototype of ultrasonic tool shank is fabricated and evaluated in experiments. It demonstrates that the dual working frequency of the ultrasonic transducer is 34.9 kHz and 104.5 kHz in the first and third mode resonant vibration, which is benefit to the low and high frequency drilling. Three working modes of the low, high, and coupling frequency vibration is measured by a self-developed ultrasonic generator. When 40 V voltage is excited, the vibration amplitudes of the low and high frequency are 17.8 µm, and 9.3 µm, respectively, which can meet with the drilling requirement. Moreover, the coupling vibration with the low and high mode are excited simultaneously, and the motion trajectory of the coupling vibration is observed in a “M” shape, which is a novel trajectory for the precision micro-nano drilling. Micro drilling experiments show that the 35 kHz ultrasonic vibration drilling can achieve better machining effect than conventional drilling (CD). The 105 kHz ultrasonic drilling outperforms the CD, 35 kHz and coupling ultrasonic drilling at cutting force reduction. The coupling ultrasonic vibration drilling achieves the lowest chipping diameter. It is to be noted that the 105 kHz-8 µm produce the fined smoothly surface. The 35 kHz, 105 kHz and coupling ultrasonic drilling can produce better surface micro morphology with less quantity and size of craters than that of CD.

Design, preparation and electromechanical characteristics analysis of piezoelectric 1-3-type composites with sandwich polymer structures

Aiming at the shortcomings of the traditional 200 kHz 1–3 piezoelectric composites with a thickness of 7.5 mm, which have a lower electromechanical coupling coefficient, lower resolution and larger acoustic impedance, a piezoelectric 1–3-type composites with sandwich polymer structures with higher electromechanical coupling coefficient and lower acoustic impedance was designed in this paper. First, the piezoelectric composites with different volume fractions of piezoelectric ceramics and silicone rubber were simulated by COMSOL software, and the optimal volume fractions of piezoelectric ceramics and silicone rubber were obtained. Second, the first and second lateral mode frequencies of the piezoelectric 1–3-type composites with sandwich polymer structures with different kerf widths were calculated. Then, based on the electromechanical equivalent circuit, the electromechanical characteristics of the piezoelectric 1–3-type composites with sandwich polymer structures were analyzed, and the equivalent performance parameters of the piezoelectric composite were obtained. Compared with other 7.5 mm thickness 1–3 piezoelectric composites, it was found that the electromechanical coupling coefficient of the piezoelectric 1–3-type composites with sandwich polymer structures is significantly increased, and the acoustic impedance is significantly decreased. Therefore, the piezoelectric 1–3-type composites with sandwich polymer structures has great potential advantages in the preparation of high-performance piezoelectric ultrasonic transducers.

Study of the Acoustic Characteristics of Suspensions Based on Glycerol and Synthetic Diamond Microparticles Using a Resonator with a Longitudinal Electric Field

The acoustic properties of suspensions based on pure glycerol and diamond powder with a particle size of 1–2 μm and different concentrations were studied using a resonator with a longitudinal electric field. A disk resonator made of langasite with round electrodes on both sides of the plate with a frequency of 4.1MHz, operating on a longitudinal acoustic wave, was completely immersed in a liquid container with the studied suspension. Based on the measured frequency dependences of the real and imaginary parts of theelectric impedance of the resonator using an equivalent electromechanical circuit, the longitudinal elastic modulus and longitudinal viscosity coefficient of the samples were determined. Comparison of the experimental dependences of the longitudinal elastic modulus, viscosity coefficient, and longitudinal acoustic wavevelocity on the volume concentration of diamond particles in the suspension with the calculated dependences demonstrated good agreement.

Systematic design and experimental realization of a radially cascaded spherical piezoelectric transducer.

As a critical component of ultrasonic vibration systems, piezoelectric transducers play an essential role in various practical application scenarios. Recent advances in spherical transducers have been widely used in underwater sound and structural health monitoring, while the cascaded spherical piezoelectric transducer with arbitrary piezoceramic shell thickness has not been investigated. Here, we propose a radially cascaded spherical piezoelectric transducer (RCSPT) and derive its electromechanical equivalent circuit with mechanical losses, dielectric losses, and load mechanical impedances. The resulting device is composed of three concentric spherical metal shells and two radially polarized spherical piezoceramic shells. The underlying physical mechanism is the inverse piezoelectric effect, which converts electrical signals into mechanical vibrations. The effects of the spherical piezoceramic shell's thickness and location on the RCSPT are studied. We also analyze the effects of mechanical losses, dielectric losses, and load mechanical impedances on the modulus of input electric impedance of the cascaded spherical transducer. The experiments are conducted to verify the electromechanical characteristics of the resulting device, which are in good agreement with the simulated results and theoretical predictions. Our methodology will offer new possibilities for designing RCSPTs and may promote applications in various fields, such as underwater acoustic detection and structural health monitoring.

Design and Analysis of Ultrasonic Composite Transducer with a Quarter-wave Taper Transition Horn

Based on the electromechanical equivalent circuit theory, equations related to the resonance frequency and the magnifying coefficient of a quarter-wave vibrator and a quarter-wave taper transition horn were deduced, respectively. A series of 3D models of ultrasonic composite transducers with various conical section length was also established. To reveal the influences of the conical section length and the prestressed bolt on the dynamic characteristics (resonance frequency, amplitude, displacement node, and the maximum equivalent stress) of the models and the design accuracy, finite element (FE) analyses were carried out. The results show that the addition of prestressed bolt increases the resonance frequency and causes the displacement node on the center axis to move towards the small cylindrical section. As the conical section length rises, the increment of resonance frequency reduces and tends to a stable value of 360 Hz while the displacement of the node on the center axis becomes lager and gradually approaches 1.5 mm. Furthermore, the amplitude of the output terminal is stable at 16.18 μm under 220 V peak-topeak (77.8 VRMS) sinusoidal potential excitation. After that, a prototype was fabricated and validated experiments were conducted. The experimental results are consistent with that of theory and simulations. It provides theoretical basis for the design and optimization of small-size, large-amplitude, and high-power composite transducers.

Theoretical modeling and experimental verification of an improved radial composite transducer consisting of a metal ring and a radially polarized piezoelectric stack

In the ring radial transducer, the wall thickness of the radially polarized piezoelectric ceramic is limited by the polarization technology and operating voltage, resulting in the limited power capacity and vibration ability of the transducer. Hence, an improved novel radial composite transducer (nRCT) is proposed in this paper, which consists of a radially polarized piezoelectric stack and a metal ring. Piezoelectric stack is used to enhance vibration and effectively solve the problem of difficult excitation caused by large wall thickness. A new electromechanical equivalent circuit model (EECM) of the nRCT in radial vibration is established, and the relationship between frequency characteristics of the nRCT and geometric size is analyzed. The finite element method (FEM) is used to carry out numerical modeling of the nRCT and the traditional radial composite transducer (tRCT), and preliminarily verify the calculation results of EECM. Compared with the tRCT, under the same electrical excitation, the equivalent electrical impedance of the nRCT designed in this paper decreases to 26%, and the radial vibration displacement increases to 142%. Finally, the nRCT and the tRCT are fabricated, and the experimental results have well verified the results of the theoretical analysis. The proposed radial piezoelectric stack model provides a new idea for the optimal design of radial vibration piezoelectric devices, which is expected to be further applied to the design of hydrophones, piezoelectric transformers, and medical ultrasound devices.

Hollow cylindrical linear stator vibrator using a traveling wave of longitudinal axisymmetric vibration mode

Ultrasonic motors (USMs) are expected to be used in special environments: high magnetic field environments and space environments, which require lightweight and multiple degrees of freedom. However, when used as linear ultrasonic motors (LUSMs), a linear guide and a preload mechanism are required, complicating the structure. In the present paper, a hollow cylindrical linear stator without an extra linear guide has been considered. The stator consists of a metal pipe and two piezoelectric (PZT) tubes installed at both ends of the metal pipe. Their connected parts are tapered for the first longitudinal axisymmetric vibration mode of the cylinder, namely L(0,1) mode excitation, and the metal pipe is subjected to radial strain. The vibration of the stator is assumed to be one-dimensional and is modeled by an electromechanical equivalent circuit. The principle that the traveling wave is formed on the metal pipe by dual-PZT-tube phase difference excitation was clarified. Finite element analysis and some measurements were conducted to confirm that the theory was consistent. The analyses and measurements were in good agreement. Therefore, the operating principle was confirmed. The results of the transport experiment showed that the average speed of the 8.5-g slider was 7.9 mm/s.

The High-Power Piezoelectric Transformer With Multiple Outputs Based on Sandwiched Piezoelectric Transducers

This article develops a high-power piezoelectric transformer based on the sandwiched piezoelectric transducer with multioutput characteristics. An advanced high-power voltage transformation technology is realized via using the superior mechanical robustness of the sandwiched piezoelectric transducers. High-power sandwiched piezoelectric transformers (s-PTs) with the unique crisscrossed structure are constructed by four sandwiched piezoelectric transducers with two modes, i.e., antiphase mode and in-phase mode, due to the coupled vibration of the central metal cuboid. Moreover, the related electromechanical equivalent circuit model (EECM) of the s-PTs is established in which two loss factors are introduced into this model, and its impedance characteristics, voltage gains, and power gains are studied as well. The accuracy of EECM is verified by the finite element method and experiments. Results reveal that s-PT has three outputs, and its voltage gains and power ratios are affected by load resistance and loss. The output power of the s-PTs can reach at least 45 W, and the s-PTs can keep the efficiency of ∼35% under high-input electric power, which provides a new idea for high-power piezoelectric transformers.

Spherical piezoelectric transducers of functionally graded materials.

In this work, a functionally graded spherical piezoelectric transducer (FG-sPET) is proposed and an accurate theoretical model is constructed, mainly composed of a three-port electromechanical equivalent circuit model (EECM). The EECM of FG-sPET can be connected to that of other vibration systems according to the boundary conditions (force and vibration velocity), making it easier to evaluate the whole mechanical vibration system. The validity of the EECM for FG-sPET is verified by comparison with other literature. The effects of geometric dimensions and non-uniform coefficients on the vibration characteristics (resonance/anti-resonance frequencies and effective electromechanical coupling coefficient) of FG-sPET are also studied, contributing to systematically evaluating the key factors determining the vibration characteristics of FG-sPET. The proposed analytical system is of excellent guidance for the structural optimization design of functionally graded piezoelectric devices.

Electromechanical Equivalent Circuit Model for Axisymmetric PMUTs With Elastic Boundary Conditions

Here we present an electromechanical lumped equivalent circuit model for the purpose of deriving the electrical input impedance and response of air-coupled axisymmetric polymer-based piezoelectric micromachined ultrasonic transducers (pMUT). In particular, this study presents the derivation and validation of polyvinylidene fluoride-co-trifluoroethylene P(VDF-TrFE) unimorph pMUT equivalent circuit models operating in the fundamental axisymmetric mode. Derivation of the equivalent circuit model is performed via the application of the Rayleigh-Ritz energy method towards determining effective pMUT radii due to an elastic boundary condition. The results of the model are shown to be in good agreement with finite element (FEM) simulations and experimental measurements of discrete pMUTs in air. In particular, the model is able to accurately predict the effective expansion in the pMUT radius due to the elastic boundary. The electromechanical model may also be used as a tool to characterize and/or evaluate the behavior of pMUT designs for ultrasound transducer applications.[2021–0156]

Multi-mode coupled vibration performance analysis of a radial-longitudinal (R-L) ultrasonic transducer.

A radial-longitudinal (R-L) ultrasonic transducer is designed by compounding a piezoelectric ceramic and an outer metal ring on the central coupling cylinder of a longitudinal cascade transducer. This design is used to realize multi-mode vibration and increase the radiation range. By applying longitudinal and radial double excitation, three coupled vibration modes of the transducer are generated in the frequency range of 15-65 kHz. The coupled vibration dominated by radial vibration is regarded as the best vibration mode of this transducer. The electromechanical equivalent circuit and the resonance frequency equation of the transducer's coupled vibration are derived by using the equivalent elastic method and one-dimensional vibration theory and verified by the finite element method and experimental method. The results show that the electrical impedance frequency curves of the transducer from the three methods are consistent. The transducer is expected to be used in ultrasonic cleaning and liquid processing applications.

Study on multi-frequency characteristics of a longitudinal ultrasonic transducer with stepped horn

In power ultrasonic and underwater acoustic fields, the ultrasonic treatment efficiency of vibration system can be improved significantly when vibration system is driven by the multi-frequency longitudinal transducer, and the performance of vibration system is determined by the multi-frequency characteristics of transducer. In this paper, using electromechanical equivalent circuit method, the multi-frequency characteristics of a longitudinal ultrasonic transducer with a stepped horn are studied for the ultrasonic vibration system optimal design. The expressions of the input impedance and the velocity amplification ratio of the transducer are obtained. The relationships between the electromechanical coupling coefficients, the velocity amplification of the multi-frequency transducer and the excitation position of the piezoelectric stack, the size of the stepped horn are analyzed. Combined with the finite element analysis of the transducer's key performance, the optimal design parameters of the multi-frequency transducer are given. Finally, the electromechanical properties of the multi-frequency transducer are tested experimentally, the theoretical analysis and finite element results are verified well.

Radial vibration analysis for functionally graded ring piezoelectric transducers based on electromechanical equivalent circuit method

Ring piezoelectric transducers (rPETs) have attracted multitudinous attentions due to their widespread applications in hydro-acoustic engineering, structural health monitoring, and energy harvesting. However, cracks are inevitably occurred by the concentrated stress at the interface between metal and piezoelectric ceramics, which will seriously affect the performance and service life of the devices. Hence, a novel functionally graded ring piezoelectric transducer (FG-rPET) has been designed and an accurate analytical system is constructed mainly including a three-port electromechanical equivalent circuit model (EECM) which can acquire exact solutions in radial vibration. In addition, the proposed three-port EECM has excellent generalizability, which is also applicable to conventional rPETs. The validity of the EECM is verified by experiments and the finite element method. The vibration characteristics (resonance/anti-resonance frequencies and effective electromechanical coupling coefficient (keff)) with fundamental and second modes for the FG-rPETs can be regulated via the inhomogeneity index and wall thickness. Markedly enhanced keff at second modes can be obtained for the FG-rPETs with specific wall thicknesses. The proposed analytical system has important guiding significance for structure optimization design of piezoelectric devices, and the designed FG-rPET is expected to break the bottleneck of mechanical properties for the conventional rPETs.

Analysis on vibration characteristics of large-size rectangular piezoelectric composite plate based on quasi-periodic phononic crystal structure

Based on the theory of composite materials and phononic crystals (PCs), a large-size rectangular piezoelectric composite plate with the quasi-periodic PC structure composed of PZT-4 and epoxy is proposed in this paper. This PC structure can suppress the transverse vibration of the piezoelectric composite plate so that the thickness mode is purer and the thickness vibration amplitude is more uniform. Firstly, the vibration of the model is analyzed theoretically, the electromechanical equivalent circuit diagram of three-dimensional coupled vibration is established, and the resonance frequency equation is derived. The effects of the length, width, and thickness of the piezoelectric composite plate at the resonant frequency are obtained by the analytical method and the finite element method, the effective electromechanical coupling coefficient is also analyzed. The results show that the resonant frequency can be changed regularly and the electromechanical conversion can be improved by adjusting the size of the rectangular piezoelectric plate. The effect of the volume fraction of the scatterer on the resonant frequency in the thickness direction is studied by the finite element method. The band gap in X and Y directions of large-size rectangular piezoelectric plate with quasi-periodic PC structures are calculated. The results show that the theoretical results are in good agreement with the simulation results. When the resonance frequency is in the band gap, the decoupling phenomenon occurs, and then the vibration mode in the thickness direction is purer.