Design optimization of wide-band Tonpilz piezoelectric. with a bending piezoelectric disk on the radiation surface

Transcription

1 Design optimization of wide-band Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface Kenji Saijyou a and Tomonao Okuyama Naval Systems Research Center, Technical R&D Institute, Ministry of Defense, Nagase, Yokosuka , Japan Received 21 November 2009; revised 9 March 2010; accepted 10 March 2010 Wide-band Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface has been proposed to improve sonar detection performance in shallow water. This transducer is driven by utilizing two vibration modes, i.e., longitudinal and bending. Consequently, to achieve a wide-band signal transmission by this transducer, the phase difference between signals, which drive the ring-stack and the bending-disk piezoelectric resonators has to be optimized. In this paper, optimization approach of this phase difference in the design process is proposed. The effectiveness of this approach was confirmed by water-pool experiments Acoustical Society of America. DOI: / PACS numbers: At, Jx, Fx, Rj EJS Pages: I. INTRODUCTION Tonpilz piezoelectric transducer with acoustic matching layers on the front mass has been widely applied to construct sonar system. 1 4 However, since the fractional bandwidth of the transducer with an acoustic matching layer is less than half-octave, two or more acoustic matching layers are necessary to broaden the bandwidth sufficiently. As the result, the size and the weight of this type of transducer become increased, which degrade the reliability of bonding layer. To overcome this limitation, Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface has been proposed. 5 This transducer has two resonance modes; one is the longitudinal mode at low frequency region, and the other is the bending mode at high frequency region. These modes are stimulated by a conventional ring-stack piezoelectric resonator 3 and a bending-disk piezoelectric resonator, separately; and wide-band signals can be obtained by controlling the phase difference between driving signals of these resonators. In other words, optimizing of the phase difference of this transducer is necessary for broadening its bandwidth. Unfortunately, the optimal phase difference depends on the frequency. In this design, the optimal phase difference leads to the maximum value of transmitted voltage response TVR. Therefore, to apply this transducer, a variable phase shifter and/or a function generator should be included in the transmission system of sonar, which forces the transmission system to a complex system. Obtaining TVR in each phase difference at each frequency is necessary to optimize the phase difference. TVR is obtained from the sound pressure calculated by finite element method FEM in the design process of this transducer. As the result, enormous FEM computational effort is necessary for optimizing the phase difference, and this is a restriction of designing this transducer structure. a Author to whom correspondence should be addressed. Electronic mail: saijyou-kenji@jcom.home.ne.jp In this paper, two expressions of sound pressure are considered to minimize the amount of computational effort, 1 when a conventional and a bending-disk resonators are driven simultaneously, the sound pressure can be expressed using two sound pressures measured by driving each resonator separately, and 2 the sound pressure in the acoustic medium is obtained by Helmholtz integral of the pressure on the radiation surface calculated by FEM model. Moreover, phase difference optimization approach in the transducer design process is proposed. In Sec. II, the characteristics of this Tonpilz piezoelectric transducer is described. In Sec. III, the phase difference optimization approach in the design process and the proposed method to calculate the transmitted sound pressure are proposed. The comparison between experimental and simulation results is presented in Sec. IV. The effectiveness of the proposed method is summarized in conclusion. FIG. 1. Structures of various Tonpilz transducers: a conventional Tonpilz piezoelectric transducer, b Tonpilz piezoelectric transducer with a bending disk on its radiation surface, and c Tonpilz piezoelectric transducer equipped with a bending piezoelectric disk on its radiation surface J. Acoust. Soc. Am , May /2010/1275/2836/11/$ Acoustical Society of America

2 FIG. 2. Frequency characteristics of the sound pressure level. Solid lines represent characteristics of the transducers; dotted and dash-dotted lines represent the frequency response of longitudinal and bending modes, respectively. a Frequency response of a typical Tonpilz piezoelectric transducer described in Fig. 1a. b Frequency response of a transducer described in Fig. 1b when the bandwidth between longitudinal and bending mode frequencies is less than 1 octave. c Frequency response of a transducer described in Fig. 1b when the bandwidth between longitudinal and bending mode frequencies is more than 1 octave. II. CHARACTERISTICS OF THE TONPILZ PIEZOELECTRIC TRANSDUCER WITH A BENDING PIEZOELECTRIC DISK FIG. 3. Frequency characteristics of the sound pressure level of a transducer described in Fig. 1c. Solid lines represent the characteristics of the transducer; dotted and dash-dotted lines represent the contribution of a ring-stack resonator and a bending-disk resonator, respectively. a Frequency response when the phase difference between both resonators is optimized over the bandwidth. b Frequency response when the phase difference is not optimized. Figure 1 shows the structure of various Tonpilz transducers. Figure 1a is a typical Tonpilz piezoelectric transducer, which is constructed of a front mass, a ring-stack piezoelectric resonator, a rear mass, and a center bolt. 3 Figure 1b is a Tonpilz piezoelectric transducer with a bending disk on its radiation surface. 6 The structure of this transducer is similar to the typical one, except it has a circular hollow about 1 mm depth in its front mass to stimulate the bending mode. Figure 2a shows the frequency characteristics of the sound pressure level SPL of a typical Tonpilz piezoelectric transducer cf. Fig. 1a, which has only one peak contributed by the longitudinal vibration mode. Alternately, the transducer shown in Fig. 1b generates longitudinal vibration mode with a resonant frequency f 1 and bending vibration mode with a resonant frequency f 2. In case that the bandwidth between longitudinal and bending mode resonant frequencies is less than 1 octave cf. Fig. 2b, the contributions of both modes can be optimized in the intermediate frequency range IF range, i.e., the vibrations stimulated by longitudinal and bending modes are in-phase. As the result, the overlapped response with broader bandwidth can be obtained. 6 However, it is difficult to achieve more than 1 octave bandwidth by this structure. The vibrations stimulated by longitudinal and bending modes are out of phase in the IF range if the bandwidth is more than 1 octave, therefore, the SPL deteriorates within the IF range cf. Fig. 2c. A disk-type piezoelectric resonator bending-disk resonator is inserted in the bending disk shown in Fig. 1c to overcome this drawback. 5 Figure 3 shows the frequency characteristics of the SPL of a transducer described in Fig. 1c. Ring-stack and bending-disk resonators are driven separately, therefore, the phase difference between the vibrations stimulated by longitudinal and bending modes should be optimized to provide an in-phase vibration cf. Fig. 3a. However, the SPL shows depression if this phase difference is not optimized cf. Fig. 3b. Consequently, optimization of the phase difference is important to design a wide-band Tonpilz piezoelectric transducer. III. OPTIMIZATION APPROACH OF THE PHASE DIFFERENCE BETWEEN DRIVING SIGNALS A. Design process of Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface The flow chart of the design process is shown in Fig. 4. Step 1 is the initialization procedure of the FEM model of J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer 2837

3 FIG. 4. Flowchart of the design process of Tonpilz piezoelectric transducer described in Fig. 1c. The design modification and the TVR calculation will be iterated until the maximum TVR satisfies target value and the optimal phase difference is about in-phase or out of phase over the design frequency band. the transducer. Step 2 is the procedure to obtain the TVR in each phase difference at each frequency by using FEM model. The optimal phase difference and maximum TVR are plotted as a function of frequency in step 3. Step 4 is the procedure where decision must be made. If the maximum TVR is inappropriate, then the design model should be modified and back to step 2. Otherwise, continue to step 5, where the optimal phase difference is about the in-phase or out of phase over the frequency band. The iteration of the design optimization will be continued until the phase difference satisfies the target value. The TVR calculation in step 2 requires enormous FEM computational effort. For example, if phase differences are varied from 0 to 360 in 1 increment at each frequency, the FEM calculation must be repeated 360 times. This enormous FEM computational effort is a restriction of optimizing the transducer structure in the design phase. The FEM computational effort should be minimized to overcome this restriction. The TVR calculation method with minimum computational effort is proposed and described in the following subsections. B. Interaction between ring-stack and bending-disk piezoelectric resonators FIG. 5. Mutual interference between ring-stack and bending-disk resonators when both are driven simultaneously. a Equivalent diagram of the TVR. When driving voltages V L and V B are equal, the TVR is defined as the ratio of the radiated sound pressure P total at the evaluation point m where is 1 m above the reference center of the transducer. The sound pressures P L and P B, which are stimulated by the ring-stack and the bending-disk resonators, are proportional to the currents I L and I B, respectively. P total is the summation of P L and P B. b Two-terminal pair network model of a driven Tonpilz piezoelectric transducer as shown in a. Z LL, Z LB =Z BL, and Z BB are the input impedance of the ring-stack resonator, the mutual impedance, and the input impedance of the bending-disk resonator, respectively. When the reciprocity theorem is applied, the voltages V L and V B are expressed by the currents I L and I B and the impedances Z LL, Z LB, Z BL, and Z BB. In this subsection, the expression of the measured sound pressure P total when the ring-stack and the bending-disk resonators are driven simultaneously is discussed. In this subsection, the sound pressure will be described as measured pressure whether the sound pressure is measured or calculated by FEM. Let P L_openB be the radiated pressure when the ringstack resonator is driven by 1 V and the terminals of bending-disk resonator are open terminals, and let P B_openL be the radiated pressure when the bending-disk resonator is driven by 1 V and the terminals of ring-stack resonator are open terminals. If P total can be described by the linear sum of P L_openB and P B_openL, only twice FEM calculations enable us to obtain P total at arbitrary phase difference. As the result, the computational effort of FEM can be drastically reduced. To obtain the expression of P total, the interaction between ringstack and bending-disk piezoelectric resonators should be considered and will be discussed in the following. When a ring-stack resonator and a bending-disk resonator are driven simultaneously, the radiated sound pressure and the TVR will be interfered; therefore, this mutual interaction should be considered in the P total calculation. The effect of this mutual interference can be formulated with the theory of two-terminal pair network. If the TVR is defined as the ratio of the radiated sound pressure at 1 m from the reference center of transducer to the driving voltage, then, the evaluation point of the radiated pressure indicated as m, hereafter is set at the 1 m above the origin of the radiation surface cf. Fig. 5a. A stress generated by a ring-stack resonator stimulates another voltage to a bending-disk resonator, and vice versa. Therefore, a Tonpilz piezoelectric transducer with a bending piezoelectric disk as shown in Fig. 5a can be expressed with a two-terminal pair network as shown in Fig. 5b in which the reciprocity theorem is applied. The left side of the network indicates the ring-stack resonator, and the right side indicates the bending-disk resonator. V L and V B are the voltage between terminals of the ring-stack and the bending-disk resonators, respectively. I L and I B are the current through the ring-stack and the bending-disk resonators, respectively. The sound pressure stimulated by the ring-stack resonator P L at the point m is proportional to I L. Also, the sound pressure stimulated by the bending-disk resonator P B at the point m is proportional to I B. The measured sound pressure P total is the summation of P L and P B. By using four parameters Z LL, Z LB, Z BL, and Z BB, then, V L and V B can be expressed as V L = Z LL I L + Z LB I B, 1 V B = Z BL I L + Z BB I B. When the terminals of the bending-disk resonator are open terminals and the ring-stack resonator is driven by V L J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer

4 =1 V, the input impedance of the ring-stack resonator Z LL and the mutual impedance Z LB =Z BL can be derived as Z LL =1/I L_openB and Z LB =V openb /I L_openB by substituting the measured values of I L =I L_openB, V B =V openb, and I B =0 to Eqs. 1 and 2. And then the measured sound pressure at the point m becomes P total = P L = P L_openB. On the other hand, when the terminals of the ring-stack resonator are open terminals and the bending-disk resonator is driven by V B =1 V, the input impedance of bending-disk resonator Z BB and the mutual impedance Z BL =Z LB can be derived as Z BB =1/I B_openL and Z BL =V openl /I B_openL by substituting the measured values of I B =I B_openL, V L =V openl, and I L =0 to Eqs. 1 and 2. In this case, the measured sound pressure at the point m is P total = P B = P B_openL. Therefore, the impedance ratios Z LB /Z LL and Z BL /Z BB are obtained as Z LB /Z LL =V openb and Z BL /Z BB =V openl, respectively. When the ring-stack and the bending-disk resonators are driven by V L =V o and V B =V o e i i is the imaginary unit, the currents I L,V o and I B,V o are expressed as I L,V o = Z BB Z LB e i Z LL Z BB Z 2 V o = V o 1 V openl e i LB Z LL 1 V openl V openb = 1 V openle i I L_openB 1 V openl V openb V o, 3 I B,V o = Z LLe i Z LB Z LL Z BB Z 2 V o = V o 1 V openb e i e i LB Z BB 1 V openl V openb = 1 V openbe i e i I B_openL 1 V openl V openb V o, 4 where is the phase difference between both resonators, hereafter. The sound pressures P L,V o and P B,V o at point m, which are stimulated by the ring-stack and the bending-disk resonators, respectively, are then derived by P L,V o I L,V o I L_openB P L_openB = 1 V openle i P L_openB 1 V openl V openb V o, P B,V o = I B,V o I B_openL P B_openL 5 = 1 V openbe i e i P B_openL 1 V openl V openb V o. 6 And the measured sound pressure P total,v o = P L,V o + P B,V o at point m can be obtained. In this way, once the complex values of V openb, V openl, P L_openB, and P B_openL are measured, P total,v o and transmitted voltage response TVR= P total,v o /V o can be obtained at any and V o. As the result, the P total,v o calculation method as previously described can be applied to obtain the TVR at any with only four FEM calculations of V openb, V openl, P L_openB, and P B_openL. Thus, the computational effort to calculate the TVR over all can be drastically reduced. C. Calculation method of transmitted sound pressure from concentric disk vibrator TVR is obtained by FEM as described in the previous subsection. The more the FEM model size is reduced, the more the computational effort of FEM is decreased. A method to reduce the FEM model size is discussed in this subsection. The conventional FEM model consists of not only the transducer structure cf. Fig. 6a, but also surrounding medium water; cf. Fig. 6b. However, the region of the surrounding medium can be treated as homogeneous. Therefore, a method to restrict the FEM model region to the transducer structure and a plane, which includes the radiation surface of the transducer is proposed to obtain the sound pressure in the acoustic medium by Helmholtz integral. This restriction of the FEM model region enables us to reduce computational effort of FEM. The detailed description of this method is discussed in the following. Let be a three-dimensional volume which is treated as homogeneous bounded on a surface S, and a point inside or on the surface of this volume is located by the radius vector r or r, respectively cf. Fig. 6c. The sound pressure pr m at a point mr m inside is described by the Helmholtz integral equation HIE 7 as pr m =SGrr m p sr n p s r Grr mds, n 7 where p s r is the pressure on the surface S, n is the inward normal vector, Grr is the free space Green function, and /n indicates the derivative with respect to the inward normal. Now let us apply the surface boundary given in Fig. 6c to the transducer structure with homogeneous model. The surface S is divided into the planar region S o, which includes the radiation surface of transducer, and the other region S as shown in Fig. 6d. r S is the radius vector of an arbitrary point on S. Given the Sommerfeld radiation condition, as let r S, p s r S =0, and p s r S / on S, then, the integral over S in Eq. 7 can be neglected. Then, pr m can be obtained by substituting p s r and p s r/n on S o into Eq. 7. In other word, the region of the FEM model becomes restricted in the transducer structure and on the planar surface S o. Moreover, when the frequency is relatively low, i.e., the wavelength of the transmitted wave is similar as or longer than the size of the transducer, the structure of the transducer can be treated as an axis-symmetry structure. In this case, the distributions of p s r and p s r/n are axissymmetry and the surface S o is described as a circular plane. Then, the normal velocity of the plane including the surface S o can be expressed as v o r = p sr i o n = v or r a 8 0 r a, on =/2 plane in a spherical-coordinate system r,, as shown in Fig. 6e. In this equation, a is the radius of S o cf. J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer 2839

5 FIG. 6. Calculation model of the radiated sound in homogeneous medium. a The transducer structure. b The surrounding medium model of the transducer and the evaluation point to calculate the TVR by FEM. In the FEM model, the medium is modeled as homogeneous. c Three-dimensional homogeneous volume bounded by a surface S. n is the inward normal vector at an arbitrary point sr on S and ds is an element region of S about sr. The sound pressure pr m at a point mr m inside can be obtained from the distribution of sound pressure p s r over S by the HIE. d, which includes the evaluation point m is bounded by S, and S is divided into the region S o, which includes the radiation surface of transducer and the other region S. When r, p s r=0, and p s r/ on S, then, pr m can be obtained merely by calculating the HIE over S o. e Circular plane S o with radius a in a sphericalcoordinate system r,,. =/2 is constant at an arbitrary point r on S o. The angle between r and r is expressed as cos =sin cos. Fig. 6e, is the fluid mass density, o =2f o with f o is the frequency. The sound pressure pr,, stimulated by v o r is independent of the angle. Therefore, pr, can be expressed as the following Rayleigh s first integral formula: pr, = i 2 o a exp ik o r r v o rrdr, 2 0 d0 r r 9 where k o = o /c is the wave number of sound, c is the sound speed in the medium, and r r is the distance between two points r=r,, and r=r,/2,. The distance r r is expressed as r r = r 2 + r 2 2rr cos, 10 where is the angle between r and r, and cos =sin cos. In this case, the radiated pressure pr, can be described as the series expansion of spherical Bessel functions and Legendre functions. The detailed calculation method of the radiated pressure from radial distribution of the surface velocity v o r is described in the Appendix. Again, once the radial distribution of the surface pressure pr is obtained by FEM, the radiated pressure pr, can be described similarly. As the result, the boundary of the FEM model can be restricted in the transducer structure and on a plane, which includes the transducer surface, therefore, the computational effort of the FEM can be drastically reduced. IV. SIMULATION AND EXPERIMENT A prototype of the wide-band Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface is designed by the proposed method in order to examine the effectiveness of the proposed method. Figure 7a shows the frequency characteristics of V openl solid line and V openb dashed line calculated by FEM when the other transducer is driven by 1 V. The frequency in this figure and the rest of the figures is normalized by f o. Figure 7b shows the frequency characteristics of P L_openB solid line; ring-stack resonator driven by V L =1 V and P B_openL dashed line; bending-disk resonator driven by V B =1 V at point m calculated by FEM. When both resonators are driven by V L =V o and V B =V o e i, the sound pressure P total,v o = P L,V o + P B,V o 11 can be obtained by the above proposed method with the calculation of V openb, V openl, P L_openB, and P B_openL. Figure 7c shows the comparison result of P total,v o and the linear sum of P L_openB and P B_openL 2840 J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer

6 (a) (c) Phase (degree) Amplitude (V) Phase (degree) SPL (db) fo VopenL VopenB Transmitting Frequency Band (2 Octaves) Frequency (Hz) 5dB 4 fo Ptotal( q, Vo) PFEM( q, Vo) Plin( q, Vo) fo Transmitting Frequency Band (2 Octaves) Frequency (Hz) 4 fo (b) SPL (db) Phase (degree) PL_openB fo 5dB PB_openL Transmitting Frequency Band (2 Octaves) Frequency (Hz) 4 fo FIG. 7. a Frequency characteristics of the open terminals voltages V openl solid line and V openb dashed line calculated by FEM when the other transducer is driven by 1 V. f o isanormalized frequency. b Frequency characteristics of the complex sound pressures P L_openB solid line and P B_openL dashed line calculated by FEM at the evaluation point m when each transducer is driven by 1 V. c Frequency characteristics of the complex sound pressures P total,v o solid line: calculated by the proposed method, P FEM,V o dashed line: calculated by FEM, and P lin,v o dash-dotted line: linear summation of P L_openB and P B_openL at the evaluation point m when both transducers are driven by 1 V at out of phase =180. The error of P total is only db, while the error of P lin is 1.6 db. The difference between P FEM,V o dashed line and P total solid line is negligibly small. Then the dashed line cannot be observed. P lin,v o = P L_openB + P B_openL e i, 12 in which the interaction between both resonators is not considered. In this figure, P FEM,V o represents the calculated sound pressure by FEM when both resonators are driven by V o =1 V, simultaneously and the phase difference is =180. The error of the SPL is defined as the root mean square rms of the difference between P eval and P FEM over the transmitting frequency band 2 octaves, where, P eval is P lin or P total. The difference between P FEM and P lin is relatively large and the error of P lin is 1.6 db. Thus, P lin is inappropriate for calculating the TVR in the design process described in Fig. 4. On the contrary, the difference between P FEM and P total is negligibly small, and the error of P total is db. The result shows that P total is appropriate for calculating the TVR in the design process of the wide-band Tonpilz piezoelectric transducer. Figure 8a shows the frequency characteristics of TVR L_openB when the ring-stack resonator is driven by 1 V and the terminals of the bending-disk resonator are open terminals. The solid line represents the FEM result and the dashed line represents the measurement result at 4 m depth. The measurement was carried out in the Naval Systems Research Center s water pool, 15 m length, 9 m width, and 8 m depth. The horizontal straight line in the figure is the target value of the TVR. Again, the error is defined as the rms of the difference between calculation and measurement results over the transmitting frequency band. The error of the TVR in Fig. 8a is 1.4 db. Figure 8b shows the frequency characteristics of V openb. Figure 8c shows the frequency characteristics of TVR B_openL when the bending-disk resonator is driven by 1 V and the terminals of the ring-stack resonator are open terminals. The error of the TVR in Fig. 8c is 5.5 db. Figure 8d shows the frequency characteristics of V openl, where the average ratio of TVR L_openB to TVR B_openL over the transmitting frequency band is 7.3 db. Therefore, when both resonators are driven simultaneously, the predicted error value can be approximated as 20 log / / / db, 13 where the former is the error contributed by TVR L_openB and the latter is that by TVR B_openL. Figure 9 is the frequency characteristics of the TVR when both resonators are driven simultaneously at =180. TVR calculated from the proposed method cf. Eq. 11 shows fairly good agreement with the measurement result, and the error of the TVR is 2.0 db, which is less than the predicted error value derived by Eq. 13, i.e., 3.1 db. By this result, the effectiveness of the proposed method is confirmed. Figure 10a shows the frequency characteristics of the TVR calculated by FEM. In Fig. 10a, the solid line is the TVR at = opt and the dashed line is the TVR at =180, where, = opt is the optimal phase difference in which the maximum TVR value can be obtained. The agreement between TVR at =180 dashed line and TVR at = opt solid line is excellent, and the difference between them is 0.3 db. Figure 10b shows the frequency characteristics of the TVR measured at 4 m depth. Again, the agreement between TVR at =180 dashed line and TVR at = opt solid line is excellent, and the difference between them is 0.3 db. These results show that the optimal phase difference is nearly out of phase, and this prototype transducer has capability to construct a sonar system with simple transmission system. Figure 11a shows the frequency characteristics of the optimal phase difference opt. The optimal phase difference is obtained as follows: J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer 2841

7 FIG. 8. Frequency characteristics of the calculated results by FEM solid line and the measurement results dashed line. a TVR L_openB the error is 1.4 db, b V openb, c TVR B_openL the error is 5.5 db, and d V openl. 1 TVR in each phase difference is measured and/or calculated at each frequency. 2 The phase difference in which the TVR shows the maximum value is searched and the result is becoming to the optimal phase difference. The incremental value of the phase difference to obtain the optimal phase difference from the measured TVR is 10, FIG. 9. Frequency characteristics of the proposed method solid line and the measurement results dashed line results of the TVR when both resonators are driven at out of phase =180. The error is 2.0 db. FIG. 10. Frequency characteristics of the maximum value at = opt solid line and that at the phase difference =180 dashed line of a TVR calculated by FEM and proposed method the error is 0.3 db and b TVR measured at 4 m depth the error is 0.3 db J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer

8 while the optimal phase difference from the calculated TVR by the proposed method is 1. The agreement between them is admirable. Figure 11b shows the frequency characteristics of the maximum TVR. The maximum TVR calculated by the proposed method shows fairly good agreement with the measurement result, and the error of the maximum TVR is 1.9 db. Moreover, the TVR is satisfied the target value over the transmitting frequency band 2 octaves. These results confirmed the effectiveness of the proposed method. V. CONCLUSIONS In this paper, an approach to optimize the phase difference between signals, which drive the ring-stack and bending-disk resonators for the wide-band Tonpilz transducer is presented. To obtain a wide-band signal transmission by the proposed transducer, the phase difference between driving signals of these resonators has to be optimized. Unfortunately, the optimal phase difference depends on the frequency. Therefore, a variable phase shifter and/or a function generator should be included in the transmission system of the sonar for applying this transducer. To realize the wideband signal by a simple transmission system, the optimal phase difference should be in-phase or out of phase, i.e., the phase difference should be optimized in the transducer design phase. Moreover, to optimize the phase difference of this transducer, the TVR should be obtained in each phase difference over the frequency band. In the design process, the TVR has been obtained from the calculation of the sound pressure by FEM, which lead to enormous computational effort of FEM. The computational effort can be reduced as follows. 1 Considering the mutual interference between both resonators. The sound pressure when they are driven simultaneously can be expressed by two sound pressures, which are measured by driving each resonator separately. 2 The sound pressure in the acoustic medium is obtained by Helmholtz integral of the pressure on the radiation surface. Then, this calculation method is applied to the phase difference optimization approach in the design process of the transducer. The effectiveness of the proposed method is demonstrated by simulations and experiments. These results are as follows. 1 The error of P total calculated by the proposed method is negligibly small, this shows that the proposed method is appropriate for estimating the TVR in the wide-band Tonpilz piezoelectric transducer design process. 2 The TVR calculated by the proposed method shows fairly good agreement with the measurement result, and the error of the TVR is less than the predicted error value. 3 The optimal phase difference of the prototype transducer FIG. 11. Frequency characteristics of the proposed method solid lines and the measurement results at 4 m depth dashed lines of a the optimal phase difference opt and b maximum TVR the error is 1.9 db. The incremental values of the phase difference used for the calculation by the proposed method and measurements are 1 and 10, respectively. is almost out of phase over the transmitting frequency band 2 octaves, this shows that the prototype transducer has capability to construct a sonar system with simple transmission system. By these results, the effectiveness of the proposed method is confirmed, and the proposed method may thus be recommended to design the wide-band Tonpilz piezoelectric transducer. ACKNOWLEDGMENTS The authors would like to thank Mr. Hiroshi Shiba and Mr. Yuta Kitamura of NEC Corporation for valuable discussions and comments on this paper. APPENDIX: SOUND PRESSURE RADIATED FROM CONCENTRIC PLANE VIBRATOR Let a concentric normal velocity distribution be expressed by Eq. 8 on =/2 plane in a spherical-coordinate system r,, as shown in Fig. 6e. Then, the sound pressure stimulated by v o r is independent of the angle. The radiated pressure pr, can be expressed as Eq. 9. By using the following series expansions J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer 2843

9 cosk o r 2 + r 2 2rr cos r 2 + r 2 2rr cos = k o 2n +1j n k o r n n k o r P n cos, sink o r 2 + r 2 2rr cos r 2 + r 2 2rr cos = k o 2n +1j n k o r j n k o r P n cos, the Green s function in Eq. 9 can be expanded as exp ik o r r = k r r o 2n +1j n k o r n n k o r + ij n k o r P n cos = ik o 2n +1j n k o r h n 2 k o r P n cos, A1 A2 A3 where j n x x is an arbitrary real value and n n x are the spherical Bessel functions of the first and the second kinds, respectively, h n 2 x is the spherical Hankel function of the second kind, P n cos is the Legendre function, and r and r are the smaller and the larger distances between r and r, respectively. By applying the following integral formula: 0 2 then 0 P n sin cos d = 0 n:odd A4 2P n 0P n cos n:even, 2 exp ik o r r d = 2ik r r o 4n +1P 2n 0 P 2n cos j 2n k o r 0 h 2n 2 k o r. A5 From Eqs. 9 and A5, sound pressure pr, is then expressed as pr, = c 4n +1P 2n 0P 2n cos 0 = c k o a 0 v o rj 2n k o r h 2n 2 k o r k o rdk o r 4n +1P 2n 0P 2n cos k o r v o rj 2n k o rh 2n 2 k o rk o rdk o r k o a +k o r r v o rj 2n k o rh 2 2n k o rk o rdk. o A6 In the above description, the sound pressure is treated in a continuous coordinate system r,,. However, a discrete coordinate system is assigned for the FEM. Therefore, the distance r is discretized to r s s=1,2,...,s max, r 1 =0, r smax =a. Through this discretization, the integrals in Eq. A6 are replaced by the following summations: pr, = c 4n +1P 2n 0P 2n cos s=1 + s eq v o r s j 2nk o r s h 2 2n k o rk o r s k o r s s max s=s eq +1 k o r s = v o r s j 2nk o rh 2 2n k o r s k o r s k o r s, 1 2 k or 2 s =1 1 2 k or s +1 r s 1 2 s s max k or smax r smax 1 s = s max. A7 where s eq is a number that satisfies r seq rr seq+1. Equation A7 has a unique finite value for an arbitrary r,, thus, if the order of summation of n and s is changed, the value of pr, is preserved. Therefore, Eq. A7 can be transformed to s eq pr, = c 4n +1P 2n 0P 2n cos v o r s s=1 j 2n k o r s h 2n 2 k o rk o r s k o r s s max + c s=s eq +1 4n +1P 2n 0P 2n cos v o r s j 2n k o rh 2n 2 k o r s k o r s k o r s. A8 Let s max be points along the line, which is constant, and the distance of each point from the origin is r s s=1,2,...,s max, r 1 =0. Then the following two vectors are derived: p s = pr 1,,pr 2,,...,pr s,,,pr smax, T, A J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer

10 v os = v o r 1,v o r 2,...,v o r s, v o r smax T, A10 p s = A ss + A ss v os, A11 where made, p s and v os are correlated mutually by and s,s elements of the matrices A ss and A ss are 4n +1P A ss s,s =c 2n 0P 2n cos j 2n k o r s h 2 2n k o r s k o r s k o r s r s r s 0 r s r s, 4n +1P A ss s,s =c 2n 0P 2n cos j 2n k o r s h 2 2n k o r s k o r s k o r s r s r s A12 0 r s r s, respectively. At r 1 =r 1 =0, h 2n 2 k o r 1 becomes infinite. However, the spherical Bessel functions j n x and n n x can be expanded as the following power series: j n x = 2x n 1 m n + m! m=0 m!2n +2m +1! x2m, n n x = 1 n+1 n 2n 2m! 2 n x m=0 m!n m! x2m + 1 n 1 m m n! m=n+1 m!2m 2n! x2m, A13 respectively, and at the value about x=0, they can be expressed as j n x = n n x = x n 2n +1!! + oxn+2, and 2n 1!! x n+1 + ox n+1. A14 Therefore, when a real value x comes to zero, j 2n x h 2n 2 xx converges to i/4n+1 and one obtains the following equation: A ss 1,1 = ic P 2n 0P 2n cos k o r 2 2. A15 Thus, when =/2 and s=s at any s=1,2,...,s max r 1 =0,r smax =a, v os and a pressure distribution vector p os =p o r 1, p o r 2,...,p o r s,, p o r smax T are correlated mutually by p os = A 0 ss + A0 ss v os, A 0 ss s,s = c ic P 2n 0 2 k o r 2 2 r s = r s =0 4n +1P 2n 0 2 j 2n k o r s h 2, 2n k o r s k o r s k o r s r s r s 0 0 r s r s 4n +1P A 0 sss,s =c 2n 0 2 j 2n k o r s h 2 2n k o r s k o r s k o r s r s r s A16 0 r s r s. J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer 2845

11 Equations A11 and A16 are the discretized expression of Eq. A8. In the case of only p os is given, v os are obtained by v os = A o ss + Ao ss 1 p os. A17 However, since the matrix A o ss +Ao ss is generally ill conditioned, it is difficult to obtain the inverse matrix A o ss +Ao ss 1 directly. Fortunately, the singular value decomposition method can be used to overcome this problem. For simplicity, s max =s max =m is assumed. Then, A o ss and A o ss are turned into square matrices and an mm matrix of A o ss =Ao ss +Ao ss can be decomposed to A o ss =UVH, where U and V are mm unitary matrices. The mm matrix is expressed as = diag 0, 1,..., q,..., m, q... m 0. A18 If the superscript H represents the conjugate transpose matrix, then V H =V 1 and U H =U 1, therefore, the inverse matrix 1 can be expressed as 1 = diag1/ 0,1/ 1,...,1/ q,...,1/ m. The value of 1/ q increases as q decreases, and if q =0, then 1/ q diverges infinitely. To avoid this divergence, a matrix 1 c is applied for filtering each 1/ q in term of 1 by zero if q is larger than a specified limit value q cut. Using the filtered inverse matrix 1 c, an regularized velocity distribution vector v os can be obtained by vˆ os = V 1 c U H p os, A19 where the hat symbol represents the regularized value. Once vˆ os is obtained from a given p os, a regularized value pˆ s can be calculated by substituting vˆ os to Eq. A11. To determine the cut-off number q cut, the reconstruction error of the sound pressure distribution p os on the vibration surface is evaluated. After forming an m length vector P os =U H p os, a regularized vector Pˆ os is obtained by the following conditions: Pˆ osq = P os q q q cut, Pˆ osq =0 q q cut, A20 with an arbitrary selected q cut. Using a regularized vector of the sound pressure distribution pˆ os =UP os, the reconstruction error Error cut influenced by filtering is obtained by FIG. 12. The reconstruction error Error cut of the sound pressure distribution p os as a function of cut-off number q cut. Error cut = p os pˆ os 100%. A21 p os Error cut is a function of q cut, and this error decreases monotonically as q cut increases as shown in Fig. 12. In this case, if q cut =56 is selected for filtering 1, the Error cut is smaller than 1%, As the result, the regularized pressure pˆ s can be obtained by pˆ s = A ss + A ss vˆ os = A ss + A ss V 1 c U H p os. A22 1 C. H. Sherman and J. L. Butler, Transducers and Arrays for Underwater Sound, 98 Springer, New York, M. Van Crombrugge and W. Thompson, Jr., Optimization of transmitting characteristics of a Tonpilz-type transducer by proper choice of impedance matching layers, J. Acoust. Soc. Am. 77, H. Bodholt, Pre-stressing a Tonpilz transducer, J. Acoust. Soc. Am. 98, T. Inoue, T. Nada, T. Tsuchiya, T. Nakanishi, T. Miyama, and M. Konno, Tonpilz piezoelectric transducers with acoustic matching plates for underwater color image transmission, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, M. Yamamoto, H. Shiba, T. Fujii, Y. Hama, T. Hoshino, and T. Inoue, Tonpilz piezoelectric transducer with a bending piezoelectric disk on the radiation surface, Jpn. J. Appl. Phys. 42, T. Fujii, H. Ishimura, M. Yamamoto, Y. Hama, and H. Kaba, Wide band Langevin type transducer with a bending disk on the radiation surface, Proceedings of the UDT 2001, Hawaii, session 6C.3. 7 E. G. Williams, Fourier Acoustics: Sound Radiation and Nearfield Acoustical Holography Academic, London, J. Acoust. Soc. Am., Vol. 127, No. 5, May 2010 K. Saijyou and T. Okuyama: Design of wide-band Tonpilz transducer