Methods of Experimental Investigation of Acoustical Interactions between Electroacoustical Transducers in Arrays

Transcription

1 Methods of Experimental Investigation of Acoustical Interactions between Electroacoustical s in Arrays Boris S. Aronov, Tetsuro Oishi and David A. Brown Department of Electrical and Computer Engineering University of Massachusetts Dartmouth, 85 Old Westport Rd., North Dartmouth, MA BTech Acoustics, 445 Wampanog Trail, Suite 5 East Providence, RI 095 Abstract. Determining acoustical loading conditions on radiators in an underwater array is of great importance to predict the performance of the array. Acoustical interactions between radiators generate a non-uniform distribution of acoustic loading over the radiating surface of the array and often cause performance degradation on the directivity patterns as well as the transmit frequency response of the array. In the electromechanical equivalent circuit model of such an array, the influence of the radiation of the neighboring radiator transducers can be considered as a mutual radiation impedance that is an intrinsic property of a given transducer and array configuration. Our goal in this project is to improve experimental techniques to determine both active and reactive parts of mutual radiation impedances. Two experimental methods are presented based on the measurements of the coupled impedance ( Z-method ) and the acousto-motive force ( V-method ). For both methods, analytical investigations based on the electromechanical equivalent circuit analysis are shown. PACS Numbers: Hz, Ar, Fx, Pf, 43.30Yj I INTRODUCTION Acoustical interactions are often observed between electroacoustical transducer elements in an underwater array. The radiations of the neighboring radiators change the acoustical loading condition of the primary radiator. As a result, a non-uniform distribution of acoustical loading is generated over the radiating surface of the array and variations in the radiating surface velocity as well as in the radiated pressure are produced across array elements []. In the electromechanical equivalent circuit model of such an array, the influence of the radiation of neighboring transducers can be considered either as a coupled

2 impedance Z ij, U j Z ij = z ij () U i or as an equivalent acousto-motive force F ij, F ij = z ij U j () where z ij is the mutual radiation impedance between the ith and jth interacting transducers, U i and U j are the surface velocities of the ith and jth transducers, respectively. z ij is generally dependent on the configuration of the transducer, on the mode of vibration, and on the geometry of the transducer in the array. For certain array configurations as shown in Figure (a) and (b), the analytical expressions describing z ij are available and found to be sufficiently accurate to predict the actual mutual radiation impedances in practical arrays [] [3] [4] [5] [6]. However, for some practical arrays that usually have complicated configurations as shown in Figure (c), the mathematical expressions do not exist and the experimental investigations are the only way to determine mutual radiation impedances. Our goal in this project is to improve experimental techniques to determine both active and reactive parts of mutual radiation impedances. Two experimental methods are presented based on the measurements of the coupled impedance ( Z-method ) and the acousto-motive force ( V-method ). For both methods, analytical investigations using the electromechanical equivalent circuit model are shown. d a d h (a) Array of piston transducers in an infinite rigid baffle. (b) Array of ring transducers in an infinite rigid cylindrical baffle. d a (c) Array of cylindrical transducers in an infinite rigid baffle. FIGURE. Various array configurations.

3 II METHODS OF MEASUREMENTS The acoustical interactions can be conveniently analyzed by representing transducers in an array in the equivalent circuit shown in Figure, in which the self and mutual radiation impedances are connected as the acoustical loads in series with the mechanical components. To determine the values of mutual radiation impedances, we must first obtain all the other circuit parameters such as the electrical and mechanical impedances, the electromechanical transformation coefficient, and the self radiation impedance. The method to obtain these parameters can be found in the literature [7] and is not described in this paper. In the following sections, two methods of measurements are described under the assumption that all the equivalent circuit parameters are known except for the mutual radiation impedances. For the sake of simplicity, an array of two transducers is considered and the two transducers are assumed to be identical, i.e., all the circuit parameters are the same. A Z-method The experimental technique commonly used to measure mutual radiation impedances is to apply voltages having equal amplitudes to two transducers in two different phase conditions (in-phase and 80-degree out-of-phase). These conditions can be easily implemented with an impedance analyzer by connecting the two transducers electrically in parallel. By switching the polarity of one of the transducers, the two phase conditions can be introduced to the input voltages, i.e., V = ±V. Then the self and mutual radiation impedances are found to be } R = R r V =V + R r V = V (3) X = X r V =V + X r V = V } R = r = R r V =V R r V = V (4) X = x = X r V =V X r V = V :n C M R m E m V S C e R e U Z z U F = z U U :n C M R m E m V S C e R e U Z U z U FIGURE. Equivalent circuit representations of an array of two transducers.

4 where R r V =±V and X r V =±V are the measured radiation resistance and reactance under the excitation condition that V = ±V. The measurement of the mutual radiation impedances described in Eq. (4) is often not sufficiently accurate. To improve the accuracy, the case is considered in which the amplitudes and phases of two input voltages are varied. Because the vibrating velocities are proportional to the input voltages, the coupled impedance can be expressed as Z =(r cos φ x sin φ) U U + j(x cos φ + r sin φ) U (5) U where φ is the phase difference between input voltages (or between vibrating surface velocities). Equating the measured coupled resistance and reactance to the real and imaginary parts of Eq. (5), the mutual radiation impedance can be found to be r =(R r cos φ + X r sin φ) U U x =(X r cos φ R r sin φ) U U. (6) For example, when φ =90orU = ju in Eq. (6), the mutual radiation impedance becomes U r = X r U. (7) U x = R r U That is, the mutual radiation resistance may be measured proportional to the measured coupled reactance and the mutual radiation reactance proportional to the measured coupled resistance. In Eq. (7), the use of R r instead of X r as in Eq. (4) to determine x is useful to increase the accuracy of the measurement because X r is often too small to be accurately measured near the resonance. It is also possible in this method to increase the accuracy of the measurement by increasing the coupled impedance, i.e., by increasing the ratio of the velocities, U / U, or the ratio of the amplitudes of input voltages, V / V.Anexampleof the experimental apparatus of the Z-method is illustrated in Figure 3. B V-method In the V-method, one of the transducers has the equivalent acousto-motive force replaced with the coupled impedance in Figure. The output voltage of one transducer driven by the acousto-motive force generated by the other transducer in the course of acoustical interactions can be found as V out. = n V in jωc S e Z (Z m + Z ) (+ Z ) (8) (Z m + Z )

5 Impedance analyzer Function Generator Power V A Phase Shifter Power FIGURE 3. impedance. Experimental apparatus of the Z-method to determine the mutual radiation where Z m is the mechanical impedance of the transducers given as Z m = R m + j(ωm /ωcm). E In the frequency range far below resonance of the measurement transducer, the approximation of Z m + Z Z is valid and Eq. (8) can be simplified as V out n V in Z at f f jωce S (Z m + Z ) r. (9) In particular, the self radiation impedance can be considered as independent of the separation distance d if the transducers are members of an array and do not locate on the edges (i.e., the symmetry of the array is valid). Then, the relative change of the mutual radiation impedance as a function of d is obtained as Z (d) Z (d 0 ) = z (d) z (d 0 ) = V out(d) V out (d 0 ) (0) where d 0 is the reference separation distance of transducer elements. An example of the experimental apparatus of the V-method is illustrated in Figure 4. III SUMMARY The methods of measurements of mutual radiation impedances were developed in theory by the equivalent circuit analysis. The Z-method was improved by introducing a certain phase shift between two transducers as well as by changing the ratio of the input voltages. The V-method is especially practicable far below resonance of the measurement transducer to measure the relative change of the mutual radiation impedance as a function of the separation distance of two transducers. The results obtained in the V-method can be scaled according to the types of operational transducers. For example, the mutual radiation impedances that may be measured with a PZT cylindrical transducer are in the range of ka < because the resonance frequency of the PZT transducer is at ka = (where a is the radius of

6 Function Generator Power V in V out Oscilloscope Pre- FIGURE 4. impedance. Experimental apparatus of the V-method to determine the mutual radiation the cylinder). For the same radius of the cylindrical transducer but made of different materials such as single crystals, the operational frequency range can coincide with the frequency range of the measured mutual radiation impedances because the resonance frequency of such a transducer is around ka 0.7. An example of preliminary experimental results is shown in Figure 5 where mutual radiation impedances in an array of three cylindrical transducers as shown in Figure (c) were measured with both methods Z-method V-method r r x r a λ FIGURE 5. Example of preliminary experimental results of measurements of mutual radiation impedances using Z- and V-methods. (After B. S. Aronov)

7 ACKNOWLEDGMENTS This work is supported in parts by ONR 3SS J. Lindberg, BTECH Acoustics, and SBIR N REFERENCES. R. S. Woollett., Trends and problems in sonar transducer design, IEEE Transactions on Ultrasonics Engineering. UE-0, R. L. Pritchard., Mutual acoustic impedance between radiators in an infinite rigid plane, J. Acoust. Soc. Am. 3(6), , B. S. Aronov., Piezoceramic Electromechanical s, written in Russian, 990, translated in English, D. H. Robey., On the radiation impedance of an array of finite cylinders, J. Acoust. Soc. Am. 7(4), , R. T. Richards., J. B. Blottman III, and B. McTaggart., Physics of array element interaction phenomena, in Power s for Sonics and Ultrasonics, Proceedings of the International Workshop, B. F. Hamonic, et al. (Eds.), Springer-Verlag, F. Pordes and C. H. Sherman., Measurement of variation of radiation resistance with separation of pairs of underwater transducers, Proceedings of the International Congress on Acoustics,, , F. V. Hunt., Electroacoustics: The Analysis of Transduction and Its Historical Background, American Institute of Physics, College Park, MD, 98.