Numerical Investigation of Acoustic Field of a Strongly Focusing Concave Transducer with Method of Matched Expansions

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2 Numerical Investigation of Acoustic Field of a Strongly Focusing Concave Transducer with Method of Matched Expansions T.V. Sinilo, O.A. Sapozhnikov Department of Acoustics, Physics Faculty, Moscow State University, Moscow9899, Russia oleg@acs366b.phys.msu.su In recent applications of ultrasound in therapeutic treatment and nondestructive testing, strongly focused beams are employed. Theoretical prediction of the corresponding pressure field is therefore of great importance. The focusing is usually achieved by means of piezoceramic concave transducers. At large focusing angles, the theory is complicated by effect of diffraction on the curved transducer surface and its edge. To solve this problem, method of matched spherical harmonics expansions is applied. The method is based on the dividing the investigated region into internal and external domains where acoustic field is represented as spherical harmonics series. The two expansions are matched on the interface of the domains. Numerical algorithm is developed to allow computing of high order cylindrical functions with necessary precision in the case of large source aperture. Numerical modeling of acoustic fields of transducers with different focusing angles (up to 90 and different conditions on the radiator edge is performed. The results are compared with the Rayleigh integral approximation, which is widely employed for plane and slightly focusing sources. The noticeable discrepancy is found that is explained by the diffraction at the curved transducer surface. Strongly focused acoustic beams are widely employed in nondestructive testing and therapeutic treatment. The focusing is usually achieved by using of concave transducers of high aperture and large focusing angle. In this work, the acoustic field radiated by a concave source is studied by method of matched expansions. This approach is based on the mathematically exact solution of the problem in Γ S Γ B C a θ Ω i Γ L r FIGURE. Geometry of the problem. f α 0 Ω e M F infinite series. It was utilized in papers, where the field of the transducer as the concave spherical bowl mounted in the infinite rigid plane was calculated. A velocity distribution along the radiating surface was supposed to be uniform. Although the method of matched expansions gives a possibility to calculate the field of the focusing transducer with the sufficiently high precision, it still has some imperfections concerned with the existence of the computer zero and computer infinity during the numerical modeling. This limits the number of the terms in the spherical harmonics series and results in limiting precision or instability. Consequently, the method may fail when calculating pressure field from sharply focused sources of large aperture. Special attention should be paid to obtain the needed accuracy. Let us discuss the geometry of the problem (see Fig.. Axisymmetric spherical bowl, Г S, of radius, a, with the center of curvature, F, mounted in an infinite rigid plane baffle, Г В, vibrates according to the law exp(-i t. In an observation point M acoustic field can be described by the Helmholtz equation p k p 0 taking into account the Sommerfeld radiation condition at infinity and boundary conditions n ik c u on the Г S, and p 0 0 p n 0 on the Г В. Here the k /c 0 is a wave number, 0 ambient density and c 0 sound velocity in the medium, u - the amplitude of the normal velocity of the radiating surface, and the normal is

3 directed inward the medium. An auxiliary hemisphere Г L of radius a centered at the point C is introduced, where the axis of symmetry intersects the plane of the baffle. This surface separates the inner domain, i, from the outer one, e. Let us denote acoustic pressure inside of these domains as p i and p e correspondingly. Then the initial problem splits into two p i k pi p i 0, ik 0 c0u, (a,b n p e k p p e 0, e 0. (a,b n B These problems are coupled by the conditions of the pressure and normal velocity continuity on the separating surface Г L : pi pe pi p e,. (3 L L n n L L The general solution of equations ( and ( can be expressed as an expansion into spherical functions. Taking into account the acoustic pressure finite at the point C and Sommerfeld radiation conditions it can be written as n S j kr p x P cos, i n n n p e n ( h kr y P cos, (4 n n n where x n and y n are coefficients determined by the boundary conditions, r and are coordinates of the spherical system with the center at the point C (see Fig., P n Legendre polynomials, j n and h are spherical Bessel and Hankel functions of the order n correspondingly. Expansion s coefficients can be found from the boundary conditions (, (, and continuity conditions (3. In the numerical simulations, number of series terms (4 has to be restricted n,,..., NB inasmuch as with increasing the order Bessel and Hankel functions become abundantly small and abundantly large correspondingly that results in loss of the algorithm stability. On the other hand, NB should be large enough to reduce the error concerned with the truncation of the series. The optimal number appears to be of the order of NB (-4 ka that depends on the specific problem and computing accuracy. To avoid working with very large and very small magnitudes of spherical functions in the case of large ka, the algorithm of consecutive renormalizing was developed. The main idea of the method is the following. During the recurrence reckoning, the functions are normalized on their magnitude in a ( n p / 0 c 0 u ,05 0, 0, FIGURE. Theoretical acoustic pressure field amplitude dependencies on the distance along the beam axis (ka 000, The amplitude, p, normalized on the value 0 c 0 u, the distance, z, on the radius of the radiator curvature, F. Solid line calculated by means of the matched expansions method, and the dashed line using the Rayleigh integral. distinctive point (for example, r a and one stores not these values as themselves but their logarithms. Every subsequent function is calculated through the previous ones taking into account the difference of the normalizing coefficients. Such procedure enlarges greatly the field of application of the matched expansions. In Fig., solid line represents the computed value of the acoustic pressure amplitude on the transducer axis versus distance from the source. For comparison, dashed line shows the result of calculations based on the Rayleigh integral. It seen that there are significant discrepancies between the two theories in the region close to the surface. This is explained by the diffraction at the transducer surface, which can be interpreted as a multiple reflections from the concave surface. Another improvement in our approach is achieved by using a nonuniform spatial location of the points where the normal velocity of radiating surface, Γ S, is set. This allows expanding the value of the focusing angle, α 0, almost up to 90 degrees. Moreover, the aforementioned modification of the method of matched expansions without the algorithm of consecutive renormalizing leads to increase of the accuracy of calculation more than 0 times comparing with odinary method offered by Coulouvrat. REFERENCES. F. Coulouvrat, J. Acoust. Soc. Am. 94(3, (993.. D. Cathignol, and O.A. Sapozhnikov, Acoust. Phys. 45(6, (999.

4 Surface Acoustic Wave Dispersive Delay Lines as Signal Generators and Compressors A. Milewski a, S. Gawor a, W. Niemyjski b, A. Arvaniti b a Tele- and Radio Research Institute, Ratuszowa, Warsaw, Poland b Telecommunications Research Institute, Poligonowa 30, Warsaw, Poland In the solution described below two interdigital DDLs are used instead of one RAC. Two DDLa were used to generate the signal and the other two to compress it. The main problem was to design the DDLs compressing the chirp signals. Usually the compressor is amplitude weighted to obtain small compressed signal sidelobes. To achieve 30dB or lower sidelobes, very high precision of the weighting is required. To compress the signal two identical amplitude weighted DDLs were used. The lines were designed so that the summarized characteristic gives the required weighting function. INTRODUCTION Surface acoustic wave (SAW dispersive delay lines (DDLs are used to generate and compress frequency modulated signals []. The simplest DDL consists of two interdigital transducers placed on the piezoelectric substrate (fig. a. The interdigital transducer converts the electric signal to surface acoustic wave and viceversa. To ensure the linear frequency modulation, at least one of the transducers must be dispersive. If long dispersion time is required reflective array compressors (RACs are used most commonly. In these devices the characteristic is shaped by the array of reflective grooves (fig. b. Design and manufacture of RACs is much more complicated in comparison with interdigital DDLs. The insertion losses of RACs are also much greater. a c V G ~ IDT transducers R L Reflectors FIGURE. Surface acoustic wave devices: a interdigital dispersive delay line, b reflective array compressor. PROPERTIES OF THE CHIRP SIGNAL Most commonly a flat envelope linear frequency modulated chirp waveforms are used. Main parameters of the waveform are: duration time T, centre frequency f 0, and frequency bandwidth B. The shape of the frequency response depends on the value of the time - bandwidth product TB. For large TB product the amplitude response is almost rectangular (fig. a. a b A [db] f [MHz] A [db] f [MHz] FIGURE. Frequency response of the linear chirp signal with TB87.5 (f 0 30MHz, T50 s, B3.75NHz: a expander, b compressor. To compress the expanded signal, the matched filter is used. Usually the amplitude weighting of the compressor must be implemented to reduce the compressed signal sidelobes (fig. b. For instance using of Hamming weighting function gives 4dB sidelobes. In practice due to amplitude ripple of the expander characteristic the sidelobes are higher especially for signals with low TB product (fig SL [db] TB FIGURE 3. Dependence of the sidelobes level on the TB product.

5 GENERATION AND COMPRESSION OF CHIRP SIGNALS USING CASCADE OF DDLS In same applications it is more convenient to use cascade of two interdigital DDLs instead of one RAC. In this solution the dispersion time of each DDL is two times shorter. Because the TB product is also two times smaller the amplitude ripple of generated signal are much greater. What is worse the signal generated by the first DDL passes through the second DDL and the ripple adds up. To show the phenomena some calculations were made for signal with following parameters: centre frequency 30MHz, dispersion time 50 s, bandwidth 3.75 s. The frequency characteristics of the signal are presented in fig.. The signal may be generated using two DDLs with 5 s dispersion time (the other parameters are the same as above. The frequency response of the signal generated using the lines is presented in fig. 4. a b a [db] t [us] FIGURE 6. Theoretical response of the compressed signal. PRACTICAL RESULTS We designed two dispersive delay lines with parameters presented in Table : Table. Parameters of the designed DDLs. Parameter Expander Compressor centre frequency dispersion time 8 7 bandwidth The lines were designed to generate and compress 50 s linear chirp signal described above. Increasing of the dispersion time made it possible to design the frequency characteristics more precisely. a [db] - - A [db] - - A [db] f [MHz] f [MHz] FIGURE 4. Frequency response of the signal generated using: a one, b two 5 s DDLs. The signal is compressed by two compressors. Each of them is amplitude weighted using square root of Hamming weighting function (fig. 5a. Both lines together give the required weighting function (fig. 5b. The compressed signal is presented in fig. 6. The increase of sidelobes level is caused by amplitude ripple of the generated signal. The amplitude ripple may be limited using additional amplitude limiter or implementing the frequency extension technique []. a A [db] - b A [db] t [us] FIGURE 7. Measured response of the compressed signal. The compressed signal is presented in fig. 7. The increase of nearest sidelobes is caused by the diffraction effect. The other sidelobes were reduced using amplitude limiter and frequency extension technique. CONCLUSION By implementing the cascade of two or more dispersive delay lines it is possible to generate and compress long chirp signals using simple interdigital devices instead of reflective array compressors. - - REFERENCES f [MHz] f [MHz] FIGURE 5. Frequency response of the compressor using : a one, b two 5 s DDLs.. H. Matthews, Surface wave filters, New York: John Willey and Sons, 977. P. Dubouis, J. P. Gragnolati, E. Psila, M. Solal, 99 Ultrasonics symposium Proc., 3-35.

6 Relation of the Insertion Loss and the Triple Transit Echo in SAW Unidirectional Transducers Jun Yamada Semiconductors & Integrated Circuits Group, Hitachi Ltd. Nippon Bildg., 6-, Otemachi -Chome, Chiyoda-ku, Tokyo, Japan Tel: Fax: yamadaju@denshi.head.hitachi.co.jp The scattering matrix of SAW unidirectional transducers is calculated on the assumption that is passive, reversible and lossless. Then the insetion loss and the triple transit echo are obtained as a function of the normalized radiation conductance. The propriety of the theoretical results is verified by experiments. INTRODUCTION Much research and development of Surface Acoustic Wave (SAW filters has been performed to obtain good performance and cost/size reduction of their electric circuits. This paper describes a simple analysis of the scattering matrix in SAW unidirectional transducers and the experimental results when external electrical load changes. A detailed knowledge of the reflection and transmission characteristics of theunidirectional transducers is necessary in the design of the low loss filters for the signal processing applications. We consider the transducers as taps whose scattering characteristics vary as a function of the electrical termination, and then obtain the insetion loss and the triple transit echo of the filter. SCATTERING MATRIX Unidirectional transducers can be indicated as the three pair ported network shown in Fig., which is identical with bidirectional transducers. In the network, an acoustic wave is incident at port, port is acoustically terminated, and port 3 is electrically terminated in load conductance Gl. An electical 90 degree phase shifter is included in the network. The scattering matrix of the transducer is expressed as (S. The matrix is symmetric and the therefore the network is reversible. S ij S ji, i,j,,3 ( The following equation would hold true the conservation theory of energy if the network is passive and lossless. (S*(S (E ( where * and (E indicate the complex conjugation and a unit matrix respecrtively. We assume for purposes of simple treatment that S ij is real. (S* (S (3 The each element of the matrix is replaced as follows. S 3 S 3 p (4 S 3 S 3 q (5 S 33 r (6 The following relations are found to be p q 4GaGl / (GaGl 4b / (b (7 r (Gl-Ga / (GaGl (b- / (b (8 where Ga, Gl and the quantity b are defined as the radiation conductance of the transducers, electrical conductance of applied load and the normalized conductance (Gl / Ga. Fig.. Schematic of an interdigital transducer with hree pair of ports. INSERTION LOSS AND TRIPLE TRANSIT ECHO The insertion loss L 3 and the triple transit echo (TTE L are defined in db by the scattering matrix of the

7 transducers. L 3-0 log (S 3 (9 L -0 log (S (0 The reflection S can be calculated by the above equations. (S (q ±p r / (p q ( Here, an index a is introduced for the directivity of the surface wave propagation. q a p ( The definition means the transducers are unidirectional or bidirectional when the index a is zero or one respectively. Thus the reflection S and transmission S 3 are expressed as a function of the index at synchronism. EXPERIMENTAL RESULTS Electrode pattern of interdigital transducer is shown in Fig.3. Apodized electrode is bidirectional transduces, which is designed for the low loss TV-IF filter application by using an impulse model. And a 8 degree rotated Y cut and X propagation LiNbO 3 is used as a piezoelectric substrate. Non-apodized transduder are four pair and three group electrodes. Apodized transducers are 50 pairs and varying pitch electrodes. A Bessel type 90 o phase shifter is used (L.8 µh and r 30 ohms in unidirectional transducers. The insertion loss and triple transit echo are measured when the source conductance Gl changes with bidirectional transducers tuned for a conjugate matched condition. The experimental results are shown in Fig.4. (S (a-r / (a (3 (S 3 4b / (a(b (4 The negative sign of the reflection r in the electrical port is elected in accordance with ref. in the crossed field model, which is true in case of bidirectional transducers. The insertion loss and the triple transit echo are obtained in Fig. as a function of normalized conductance for shunt resonant electrical load when the index changes. Fig.3 Electrode pattern of interdigital transducers. Fig. 4 Measured data and calculated curves of insertion loss and triple transit echo. Fig.. Acoustic reflection and conversion loss as a function of normalized conductance. Caluculated results are put in order as follows. ( a 0 (unidirectional (S (b- / (b (5 (S 3 4b / (b (6 ( a (bidirectional (S / (b (7 (S 3 b / (b (8 CONCLUSION The reflection and conversion losses of SAW unidirectional transducers are obtained as a function of the normalized radiation conductance from the scattering matrix, which is calculated on the assumption that it is passive, reversible and lossless. Then the propriety of the theoretical results is verified by measuring the insertion loss and triple transit echo when the electrical load changes. REFERENCES. K.Yamanouchi, N.Nyffeler and.shibayama:proc. IEEE Ultrasonics Symposium (975, p.37.. W.R.Smith,H.M.Gerard,J.H.Collins,T.M.Reeder and H.J.Shaw:IEEE Trans.,MTT-7(

8 Estimation of Equivalent Circuit Parameters for Electroacoustic Transducer Using Frequency Domain Least Square Method B. D. JUN a, N. J. CHOI b, Y. J. KIM b, J. S. LIM c and K. M. SUNG b, a LG Innotek Co., Ltd., 48- Mabuk Gusung Yongin Kyunggi,449-90, Korea b School of Electrical Engineering, Seoul National University, Seoul, 5-74, Korea c Department of Electronics Engineering, Sejong University, Seoul, , Korea In this paper, we propose a new estimation method for equivalent circuit parameters of electroacoustic transducers. This method divides the impedance equation of the equivalent circuit into two parts as electrical and mechanical part. Each part is modeled with poles and zeros in frequency domain. Least square method can be iteratively applied with the measured electrical impedance data over the desired frequency band for these poles and zeros. The impedance characteristics of electroacoustic transducer with estimated parameters are in good agreement with measured ones. INTRODUCTION In estimating the equivalent circuit parameters for the electroacoustic transducer, the resonance method has been frequently used. Sometimes, however, it is difficult to estimate the equivalent circuit parameters for the electroacoustic transducer that has complex structure. In this paper, we propose a new estimation method for equivalent circuit parameters of electroacoustic transducers. This method divides the impedance equation of the equivalent circuit into two parts as electrical and mechanical part. Each part is modeled with poles and zeros in frequency domain. Least square method can be iteratively applied with the measured electrical impedance data over the desired frequency band for these poles and zeros. The procedure is as follows: At the first step the impedance equation for the electrical part are estimated with the measured impedance data by least square method. Next, the impedance equation for the mechanical part is estimated with the difference between measured impedance data and the estimated impedance data from the previous result. Finally the estimated impedance data from the above results is compared with the measured impedance data. These three steps are iterated until the error is considerably small in final step. MEASUREMENT AND ESTIMATION OF THE EQUIVALENT CIRCUIT PARAMETERS FIGURE. demonstrates the structure of the electromagnetic transducer. FIGURE. Structure of the electroacoustic transducer Since the diaphragm is concave and since the forces are applied around its perimeter, it behaves like a suspended rigid piston, or, in other words, like a mechanical resonator. Both E shaped magnet and the coil are annular. FIGURE. represents the equivalent circuit of the electroacoustic transducer.

9 Z in r 0 r b Z e l b l0 r Z m l c The electrical input impedances are measured over the effective frequency band. Using least square method with equation (6 and measured data, equivalent circuit parameters are derived. Electrical input impedance plots of eletroacoustic transducer with measured data and with equivalent circuit parameters are shown in FIGURE 3. FIGURE. The equivalent curcuit of the eletroacoustic transducer The electrical impedance of the circuit representing electroacoustic transducer is written Z Z Z in ( e m Impedance Magnitude(Ohm where Z e and Z m are the impedances for the electrical part and mechanical part respectively. Z e Z m r r 0 b jω ( r l r l r l ( jω b b b jωb jω b a jωa r r jωl jωrl jωc r jωl jωl jωβ α jωα 0 ( jω α 3 ( jω l0lb rlc ( (3 From equation ( and equation (3, the equivalent circuit parameters can be derived. r 0 a, l 0 a r α, l α, c α, r b b 3 a, l α α b b 3 a, (4 Phase(Deg Frequency(Hz FIGURE 3. Electrical input impedance plots of the elctro acoustic transducer (solid line : measured data, dotted line : equivalent circuit data CONCLUSION By using the measured impedance data and a sequential computer program, the equivalent circuit parameters of the elctroacoustic transducer can be derived with considerably small error. The impedance characteristics of elctroacoustic transducer with estimated parameters are in good agreement with measured ones. Let the polynomial ratio Z B( jω b b jω LL b n ( jω ( jω n ( jω n A( jω a a jω LL an (5 REFERENCES. Acoustic and Electronics, Artech House Mario Rossi, 988. pp. 339~347. denote the impedance function to estimate the experimental data Z(jω known at the frequency ω k. A natural goal of optimization is to minimize the error. min a, b A( jω Z( jω B( jω k (6. Rolf Johansson, System Modeling and identification, Pretice-Hall, Englewood Cliffs, New Jersey, 993. pp. 90~9.

10 Recent Advances In -3 Piezoelectric Polymer Composite Transducer Technology For Auv/Uuv Acoustic Imaging Applications K. C. Benjamin NAVSEA Division Newport RI USA 084 Abstract: Over the past five years, the use of -3 piezoelectric polymer composite has been studied under various U.S. Navy funded transducer research programs. As a transduction material, the -3 piezoelectric composite offers many advantages for AUV/UUV transducer designers. Broad bandwidth, high transmit / receive response, low cost of fabrication, mechanical ruggedness, and the ability to form conformal shapes, make this material both a unique and intriguing medium for the sonar transducer designer. The -3 piezocomposite panels are comprised of several piezoceramic rods aligned vertically through the panel s thickness with each rod surrounded with a specific polymer epoxy. Volume fractions of ceramic to polymer can vary up to 50% with panel thickness ranging between fractions of a millimeter (-4 MHz to 5 millimeters (50 70 khz. This presentation will elucidate the application of this unique material to a variety of practical transducer designs and describe some of the recent device demonstrations along with measured performance data. [Work sponsored by the US Navy]. INTRODUCTION The initial demonstration of the -3 piezocomposite panel as a viable transduction medium [] was conducted at the Materials Research Laboratory (MRL of Pennsylvania State University around 977 under 6. funding support from the Office of Naval Research (ONR. The active panels were constructed primarily by dicing and backfilling solid ceramic blocks, making the -3 composite technology well suited for small, high-frequency (> MHz biomedical transducer arrays []. Thus, the first commercialization of the material by the biomedical imaging community began in early 980. In early 990, [3] Materials Systems Incorporated (MSI perfected a ceramic injection molding process by mixing ceramic powder with a wax binder. Similar to injection molding of plastics, the process enabled larger -3 piezocomposite panels to be formed more cost effectively. Throughout the 990s, MSI s injection-molded -3 piezoceramic has steadily gained acceptance as a viable transducer material in the U.S. and U.K. Navies, as well as in the commercial sectors in the United States, Japan, and Europe. was the focus of this study. The sensors employed a -3 piezoceramic-thermoplastic composite that is conformable at elevated temperatures and capable of deep ocean operation. The thermoplastic epoxy phase of the composite allows the active panel to be shaped to any reasonable specific vehicle geometry. For the current receiver application, low-profile (<.0 mm coaxial cables embedded within the composite panels provide addressing between the acoustic array elements and the processing electronics. The array fabrication approach is applicable for both transmit and receive operation. Figures and show an example of one two-dimensional array aperture before and after curving. Figure 3 shows this array (with black window mounted on the forward section of a vehicle along with other similar demonstration arrays. Doubly Curved Two-Dimensional Conformal Array Development This section describes a hardware demonstration that illuminates the unique advantages -3 piezocomposite affords in the fabrication of twodimensional arrays. The development of doubly curved acoustic arrays for unmanned undersea vehicle (UUV applications FIGURE. Doubly curved two-dimensional UUV array (before curving SESSIONS

11 operational bandwidth and spatial fidelity typical of -3 piezocomposite constructs. FIGURE. Doubly curved two-dimensional UUV array (after curving FIGURE 4. Measured receiving voltage response for a single array element within the large flank array shown in figure 3 REFERENCES. R. E. Newnham, D. P. Skinner, and L. E. Cross, Connectivity and Piezoelectric-Pyroelectric Composites, Materials Research Bulletin, vol. 3, 978, pp W. A. Smith, A. Shaulov, and B. A. Auld, Tailoring the Properties of -3 Composite Piezoelectric Materials for Medical Ultrasonic Transducers, FIGURE 3. Conformal two-dimensional UUV array suite during testing Figure 4 corresponds to a measured receiving voltage response in db re volt per mpa for a typical array element within the larger flank array shown in figure 3. The frequency parameter spans the range 0 to 00 khz. Note the well-behaved broadband response that the 3 piezocomposite construct provides. Proceedings of 985 IEEE Ultrasonic Symposium, 985, pp L. Bowen and K. French, Fabrication of piezoelectric/polymer composites molding, International Proceedings Symposium by injection 8th IEEE of the on Applications of Ferroelectrics, 99, pp CONCLUSIONS The -3 piezoelectric composites enable a variety of unique transducer and array designs to be realized for underwater applications. The prototype doubly curved arrays, briefly described in this article, demonstrate the wide SESSIONS

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