Study on the compound multifrequency ultrasonic transducer in flexural vibration

A new type of compound multifrequency ultrasonic transducer is analyzed in this paper. The compound multifrequency ultrasonic transducer consists of two sandwiched ultrasonic transducers and a rectangular radiator. In virtue of the coupling between longitudinal vibration of the sandwiched ultrasonic transducers and flexural vibration of the rectangular radiator, the compound multifrequency ultrasonic transducer can produce several resonance frequencies. Some compound multifrequency ultrasonic transducers are designed and simulated by finite element method (ANSYS), and modal shapes and harmonic response are analyzed. The compound multifrequency ultrasonic transducers are designed and manufactured. The resonance frequencies are measured and compared with the numerical results. The effect of the geometrical dimensions of the compound multifrequency ultrasonic transducer and the location of two sandwiched ultrasonic transducers on the compound multifrequency ultrasonic transducer is discussed. It is shown experimentally and numerically that the compound multifrequency ultrasonic transducer has several resonance frequencies.     

̀Spatial impulse response of a rectangular double curved transducer

Calculation of the pressure field from transducers with both a convex and a concave surface geometry is a complicated assignment that often is accomplished by subdividing the transducer surface into smaller flat elements of which the spatial impulse response is known. This method is often applied to curved transducers because an analytical solution is unknown. In this work a semi-analytical algorithm for the exact solution to a first order in diffraction effect of the spatial impulse response of rectangular-shaped double curved transducers is presented. The solution and an approximation to it are investigated. The approximation reformulates the solution to an analytically integrable expression, which is computationally efficient to solve. Simulation results are compared to FIELD II simulations. Calculating the response from 200 different points yields a mean error for the different approximations ranging from 0.03% to 0.8% relative to a numerical solution for the spatial impulse response. It is also shown that the presented algorithm gives consistent results with FIELD II for a linear flat, a linear focused, and a convex nonfocused element. The solution involved a three-point Taylor expansion and gave an accuracy of 0.01%.     

Diffraction phenomena in the field of a circular piezoelectric transducer viewed in the time domain

Transient diffraction phenomena in the field of a piston transducer have been explained theoretically in terms of the direct and edge wave contributions to the piston response. Previous experimental studies by other workers have verified this concept for single cycles of 4 MHz radiation. Faster transients are difficult to study using thin piezoelectric transducers due to transducer resonance. Thick piezoelectric transducers of very low resonant frequency can be used to generate and receive step function acoustic transients with rise times of 20 ns or less and the problems of transducer resonance can be avoided. Experimental results are presented which show that when such a step function is propagated between two thick transducers the time course of the received wave can be explained in terms of the direct wave/edge wave model in conjunction with the theory of transient reverberations in piezoelectric elements.     

Piezoelectric transducer parameter selection for exciting a single mode from multiple modes of Lamb waves

Excitation and propagation of Lamb waves by using rectangular and circular piezoelectric transducers surface-bonded to an isotropic plate are investigated in this work. Analytical stain wave solutions are derived for the two transducer shapes, giving the responses of these transducers in Lamb wave fields. The analytical study is supported by a numerical simulation using the finite element method. Symmetric and antisymmetric components in the wave propagation responses are inspected in detail with respect to test parameters such as the transducer geometry, the length and the excitation frequency. By placing only one piezoelectric transducer on the top or the bottom surface of the plate and weakening the strength of one mode while enhancing the strength of the other modes to find the centre frequency, with which the peak wave amplitude ratio between the S0 and A0 modes is maximum, a single mode excitation from the multiple modes of the Lamb waves can be achieved approximately. Experimental data are presented to show the validity of the analyses. The results are used to optimize the Lamb wave detection system.     

Acoustical fields from rectangular ultrasonic transducers for non-destructive testing and medical diagnosis

Theoretical results are given for pressure and intensity in the acoustical field produced by rectangular ultrasonic transducers. Comparison between the fields of square, circular and rectangular transducers is made showing that the field of a rectangular transducer, by comparison with that produced by square and circular transducers, may be divided into three zones rather than two: the very nearfield, the nearfield and the farfield. Some experimental results are shown confirming the general shape of the predicted curves. Emphasis is put on the difficulty of interpreting quantitative ultrasonic results with rectangular probes.     

Acoustic field modeling for physiotherapy ultrasound applicators by using approximated functions of measured non-uniform radiation distributions

The strongest therapeutic effects in ultrasonic physiotherapy are mainly produced at the first centimeters, i.e. close to the applicator surface and, in general, only in the near-field zone. The acoustic field produced in practice by this type of transducers differs from the classical models because the vibration distribution on the real transducer surfaces is non-uniform. However, neither models using uniform distribution, nor those using typical non-uniform distribution patterns for the source accurately represent the radiation of this kind of transducers. Although this therapy is widely used and many efforts have been made in experimentally studying the patterns of ultrasound radiation produced during physiotherapy applications (IEC-61689, 1998), additional modeling researches still would be needed in order to achieve improved models giving field patterns closer to the measured ultrasonic results. In this paper, acoustic patterns produced from two source radiation functions are proposed and evaluated for field modeling of physiotherapy applicators. Both the functions are approximations to the pressure distribution measured close to the emitting surface and they are based on the modulation of the classical simply-supported function using either sinusoidal or Bessel-type distributions. The simply-supported function is accounted for the radiator-fixing condition and the modulation function simulates the complex vibration distribution of this kind of transducer. The modulator Bessel function is based on reports about Bessel-type vibration distributions found in piezoelectric disk resonators. The use of a selected sinusoidal segment represents another analytical option for obtaining an approximated behavior of the measured data in a real applicator. Both the field models are implemented using the finite element method (FEM) to obtain the numerical solution of wave equation at each point in the radiated space. The solution is reached by considering axisymmetric radiation in attenuation-free media. The results indicate the viability of applying an adequate model for acoustic field calculation by simulating the radiating distribution on the emitting surface as either sinusoidal or Bessel-modulated functions. Models using both the functions describe reasonably real behaviors, but those based on Bessel functions are better correlated with the measurements. The results for three commercial applicators indicate the possibility of representing, with adequate verisimilitude, the acoustic field radiated by physiotherapy ultrasound transducers using linear combinations of Bessel profiles describing the radiation source.     

Ultrasonic field modeling for immersed components using Gaussian beam superposition

The Gaussian beam (GB) superposition approach can be applied to model ultrasound propagation in complex-structured materials and components. In this article, progress made in extending and applying the Gaussian beam superposition technique to model the beam fields generated by transducers with flat and focused rectangular apertures as well as with circular focused apertures is addressed. The refraction of transducer beam fields through curved surfaces is illustrated by calculation results for beam fields generated in curved components during immersion testing. In particular, the following developments are put forward: (i) the use of individually determined sets of GBs to model transducer beam fields with a number of less than ten beams; (ii) the application of the GB representation of rectangular transducers to focusing probes, as well as to the problem of transmission through interfaces; and (iii) computationally efficient transient modeling by superposition of ‘temporally limited’ GBs.     

Extension of the distributed point source method for ultrasonic field modeling

The distributed point source method (DPSM) was recently proposed for ultrasonic field modeling and other applications. This method uses distributed point sources, placed slightly behind transducer surface, to model the ultrasound field. The acoustic strength of each point source is obtained through matrix inversion that requires the number of target points on the transducer surface to be equal to the number of point sources. In this work, DPSM was extended and further developed to overcome the limitations of the original method and provide a solid mathematical explanation of the physical principle behind the method. With the extension, the acoustic strength of the point sources was calculated as the solution to the least squares minimization problem instead of using direct matrix inversion. As numerical examples, the ultrasound fields of circular and rectangular transducers were calculated using the extended and original DPSMs which were then systematically compared with the results calculated using the theoretical solution and the exact spatial impulse response method. The numerical results showed the extended method can model ultrasonic fields accurately without the scaling step required by the original method. The extended method has potential applications in ultrasonic field modeling, tissue characterization, nondestructive testing, and ultrasound system optimization.     

A new type of high power composite ultrasonic transducer

A new type of high power composite ultrasonic transducer was proposed and studied. The composite transducer consists of a sandwich longitudinal piezoelectric transducer, an isotropic metal hollow cylinder with large radial dimension, and the front and back metal radiation mass. By means of its special structure and Poisson’s effect, the composite transducer can produce vibrations both in its longitudinal and radial directions, and therefore, it can radiate sound waves in three-dimensional space. The electro-mechanical equivalent circuit of the composite transducer was derived and the resonance frequency equation was obtained analytically. Numerical methods were used to simulate the vibration of the composite transducer, and the vibrational displacement distribution, the resonance frequency and the radiation sound field are given. Some composite transducers are designed and manufactured; their resonance frequencies and the radiation acoustic field are measured and compared with the analytical and numerical results. It can be seen that the measured frequencies and acoustic field contour are in good agreement with the analytical and numerical results. It is expected that this kind of composite ultrasonic transducer can be used in more and more power ultrasonic applications, such as ultrasonic cleaning, ultrasonic extraction, ultrasonic sonochemistry and other ultrasonic liquid processing, where high ultrasonic power and large ultrasonic processing space are needed increasingly.     

Exact solution of resonant modes in a rectangular resonator

The resonant modes of rectangular two-dimensional optical resonators were analyzed exactly. Based on the characteristics of the Bessel function, the resonant frequencies of the rectangular microcavities are expressed in a simple way. In addition, a simple rule to judge when the finite length of a rectangular resonator can be considered infinite is given in realistic applications. The solution that is presented should be useful in possible applications of the rectangular resonators as filters for dense wavelength-division multiplexing.     

The radiation impedance of a rectangular piston

The radiation impedance of a rectangular piston is expressed as the Fourier transform of its impulse response, which is obtained from the recent work of Lindermann [1]. The analytical evaluation of the transform is performed and new integral expressions are presented for both the radiation resistance and reactance. The integrals are readily evaluated in terms of elementary functions at both the low and high frequency limits. The integrals are also expressed as series of Bessel functions which are valid for all frequencies and aspect ratios. Numerical results are presented to illustrate the behavior of the radiation resistance and reactance as a function of the aspect ratio of the piston and a normalized frequency parameter. Additional numerical results are then presented to illustrate the accuracy of the analytical expressions for the radiation resistance and reactance at low and high frequencies. Finally, numerical results are presented to illustrate the application and accuracy of using standard FFT algorithms to evaluate the radiation resistance and reactance directly from the impulse responses.     

Wave guide imaging through Time Domain Topological Energy

Time Domain Topological Energy (TDTE), uses the reflected ultrasonic field recorded by an array of transducers placed on the boundary of the inspected medium. Two numerical determinations (forward and adjoint problems) of the acoustical field inside a reference medium are necessary to compute an image. Topological Energy is defined as a variation of topological sensitivity or gradient and comes from the field of mathematical optimisation. Recent developments for Non-Destructive Testing have shown the analogy with Time Reversal concepts. Time Reversal mirrors have been employed for various applications in a wide number of situations including wave guides where very good re-focalization performances have been obtained with a reduced number of transducers instead of an array of transducers. Moreover recent works have enlightened that the reverberation properties of a wave guide allow to re-focalize using Time Reversal with only one transducer. For TDTE imaging a single transducer placed at one end of a wave guide has been modelled. The boundaries of the wave guide create virtual sources that can be understood as a virtual array of transducers. Numerical and experimental results are presented using TDTE and a single transducer in a wave guide for targets and conditions of increasing complexity.     

A new algorithm for spatial impulse response of rectangular planar transducers

Previous solutions for spatial impulse responses of rectangular planar transducers require either approximations or complex geometrical considerations. This paper describes a new, simplified and exact solution using only trigonometric functions and simple set operations. This solution, which can be numerically implemented with a straightforward algorithm, is an exact implementation of the Rayleigh integral without any far field or paraxial approximation. Additionally, a nonlinear relationship was also established for spatial impulse responses from two field points which share the same projection point on the transducer surface plane. By incorporating this relationship in the algorithm, the computational efficiency of spatial impulse responses and continuous fields is improved about 20-folds and 14-folds, respectively. This algorithm has practical applications in designing l-D linear/phased arrays, 1.5-D arrays and 2-D arrays, as demonstrated through numerical simulations with array transducers. Experiments were also conducted to verify the new solution and results show that the algorithm is both accurate and efficient. The application of this method may include development of ultrasound imaging system for hard and soft tissue nondestructive assessment.     

An ultrasonic measurement model using a multi-Gaussian beam model for a rectangular transducer

To date, ultrasonic measurement models have primarily treated systems where circular transducers are used. Recently, however, a highly efficient ultrasonic beam model for a rectangular transducer has also become available where the transducer is represented as a superposition of a relatively few Gaussian beams. Thus, using the multi-Gaussian beams, we developed ultrasonic measurement models for systems where a rectangular transducer is employed. In this paper, we describe the developed models including the beam model, the efficiency factor for a rectangular transducer and far-field scattering models for some standard scatterers. Furthermore, the accuracy of the proposed model is verified by the comparison of the model-based predictions to the experimental measurements.     

Finite amplitude pressure field of elliptical and rhomboid transducers in three dimensions

Design of different type of transducers to enhance image quality by forming narrow beams at the principals of nonlinear acoustics is considered in the paper. Thus, the nonlinear pressure fields of elliptical and rhomboid transducers were simulated in three dimensions. The simulation method presented in this study is based on Aanonsen’s model for circular sources, and closely follows the model that recently explored for the nonlinear wave propagation due to square and rectangular sources in three dimensions [Kaya et al. “Pressure field of rectangular transducers at finite amplitude in three dimensions,” Ultrasound in Med. Biol., vol. 32, no. 2, pp. 271–280, 2006]. It is assumed that elliptical and rhomboid sources are plane sources, and driven at 2.25 MHz fundamental frequency. Typical results of nonlinear acoustical pressure field simulation are presented there in three dimensions for elliptical and rhomboid sources and compared with the results for rectangular source. The similarities and differences between the nonlinear pressure field of rectangular, elliptical and rhomboid sources are discussed. The numerical results show that diffraction effects and acoustical beam cross section depend on the source geometry a lot. It is noticeable that the nonlinear pressure field of a rectangular source has a broader beam profile than elliptical and rhomboid source.     

双激励全波长压电陶瓷超声换能器工作特性分析

对由两个半波长夹心式压电陶瓷换能器级联而成的双激励全波长压电陶瓷超声换能器进行了研究,给出了这种换能器优值的相关参量表达式,通过数值计算分析了该换能器的结构和材料参数对其特性的影响。研究结果表明,当两组压电陶瓷堆分别位于各自所在半波长换能器的位移节点,或者在压电陶瓷片数量确定情况下两组压电陶瓷堆的片数相同时,换能器均能得到最大的优值;在相当大的范围内增加压电陶瓷片的数量,换能器的优值稍有降低,但力因子迅速增大;金属块材料对换能器的优值影响甚微。可见双激励全波长换能器能在不明显降低换能器综合性能的情况下有效增加压电陶瓷的体积,因而可大幅提高换能器的功率容量和负载能力,更适用于大功率重负载的场合。      

Operating characteristics analysis of a dual-excited full-wavelength piezoceramic ultrasonic transducer

A dual-excited full-wavelength piezoceramic ultrasonic transducer as a cascade of two half-wavelength sandwich piezoceramic transducers is studied.The relevant parameters' expressions of the figure of merit N for the transducer are derived,and the effects of the structure and material parameters of the transducer on its characteristics are further analyzed by numerical calculation.The results show that when the two piezoceramic stacks are respectively located at the displacement nodes of their own half-wavelength transducers,or the two piezoceramic stacks have the same number of pieces in the case of a certain number of piezoceramic pieces,the figure of merit of the transducer can reach a maximum.With increasing of the number of piezoceramic pieces in a fairly large range,the figure of merit of the transducer slightly decreases,but the force factor of the transducer increases rapidly.The metal materials of the transducer have little effect on its figure of merit.Thus it can be seen that the dual-excited full-wavelength transducer can effectively increase the volumes of the piezoceramic stacks in the case of that the transducer's comprehensive performance has only a little bit of degradation,so it's power capacity and load capability can be dramatically improved,which means the transducer is more suitable for high power and heavy load applications.     

Modeling the quasicompressional wave field of a rectangular transducer in a transversely isotropic solid

A fast model for simulating the transient quasicompressional wave field of a rectangular ultrasonic transducer directly coupled to a transversely isotropic elastic half-space of general orientation is developed. The so-called two-tier asymptotic approach and the uniform stationary phase method are used to derive the high-frequency asymptotics of time-harmonic displacements. Then, transient fields are modeled by means of harmonic synthesis. In geometrical regions, the formulas involve elementary and inside boundary layers, well-known special functions (Fresnel integral and generalized Fresnel integral), and are applicable in the radiating near field. The asymptotics elucidate the physics in terms of various arrivals and give explicit dependence of the radiated waves upon model parameters. The asymptotic code is tested against a direct numerical solution. It is at least a thousand times faster but describes accurately both arrival times and amplitudes of various pulses radiated by the transducer.     

Elastic guided wave propagation in a periodic array of multi-layered piezoelectric plates with finite cross-sections

In this study, we present a model study of guided wave dispersion and resonance behavior of an array of piezoelectric plates with arbitrary cross-sections. The objective of this work is to analyze the influence of the geometry of an element of a 1D-array ultrasound transducer on generating multi-resonance frequency so as to increase the frequency bandwidth of the transducers. A semi-analytical finite-element (SAFE) method is used to model guided wave propagation in multi-layered 1D-array ultrasound transducers. Each element of the array is composed of LiNbO₃ piezoelectric material with rectangular or subdiced cross-section. Four-node bilinear finite-elements have been used to discretize the cross-section of the transducer. Dispersion curves showing the dependence of phase and group velocities on the frequency, and mode shapes of propagating modes were obtained for different geometry consurations. A parametric analysis was carried out to determine the effect of the aspect ratio, subdicing, inversion layer and matching layers on the vibrational behavior of 1D-array ultrasound transducers. It was found that the geometry with subdiced cross-section causes more vibration modes compared with the rectangular section. Modal analysis showed that the additional modes correspond to lateral modes of the piezoelectric subdiced section. In addition, some modes have strong normal displacements, which may influence the bandwidth and the pressure field in front of the transducer. In addition, the dispersion curves reveal strong coupling between waveguide modes due to the anisotropy of the piezoelectric crystal. The effect of the matching layers was to cluster extensional and flexural modes within a certain frequency range. Finally, inversion layer is found to have a minor effect on the dispersion curves. This analysis may provide a means to analyze and understand the dynamic response of 1D-array ultrasound transducers.