Equivalent circuit for broadband underwate transducer

0885–3010/$25.00 © 2008 IEEE 2079IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, vol. 55, no. 9, SEPTEMBER 2008 Correspondence Equivalent Circuit for Broadband Underwater Transducers R. Ramesh and D. D. Ebenezer Abstract—A method is presented to determine the equiva- lent circuits of broadband transducers with 2 resonances in the frequency band of interest. The circuit parameters are refined by least-squares fitting the measured electrical conductance data with this model. The method is illustrated by computing the conductance and susceptance of the equivalent circuits of 3 types of broadband transducers and comparing them with the measured values. The equivalent circuit of a transducer is necessary for designing filters that match the impedances of the transducer and the power amplifier that drives the trans- ducer. I. Introduction Broadband transducers that have nearly flat and high transmitting voltage response (TVR) over one octave or even more have been developed in recent years [1]. However, there is considerable variation in the imped- ance of the transducer in the band in which the TVR is nearly flat. It has been a general practice to use a tuning coil (inductor) connected in parallel with the transducer to make the impedance, as seen by the power amplifier, nearly resistive. However, this approach is suitable only when there is just one resonance in the impedance, in the band of interest. The impedances of broadband projectors developed us- ing piezoelectric ceramics and piezocomposite transducers, generally, have more than one resonance in the impedance in the band in which the TVR is nearly flat. Therefore, there has been recent interest in finding a matching cir- cuit that will serve as an interface between the power am- plifier and such transducers to improve the power transfer characteristics [2]. Analysis of the transducer, the matching filter, and the power amplifier is considerably simplified if the transduc- er is represented by an equivalent circuit [3]. It is further simplified if the values of the components of the equiva- lent circuit of the transducer are lumped (components are pure inductors, capacitors, or resistors) and independent of frequency—as is the case with electrical components. Several authors have studied the electrical equivalent circuits of different types of transducers. For example, Shuyu [4] has studied the equivalent circuit of an ultra- sonic transducer consisting of a flexural plate excited by a longitudinal PZT driver for air-coupled ultrasonic ap- plications. Marshall and Brigham [5] have estimated the equivalent circuit parameters of transducers with a low figure of merit. They have determined 4 parameters of the equivalent circuit from the analysis of capacitance curve. Sherrit et al. [6] have proposed an equivalent circuit model of a piezoelectric thickness-mode vibrator and estimated the circuit parameters from complex material coefficients. Church and Pincock [7] have described a method to deter- mine the equivalent circuit parameters of small acoustic transmitters at or near resonance. However, these analy- ses are restricted to transducers with one resonance in the operating band of interest. Coates and Maguire [8] have derived approximate expressions of equivalent circuit pa- rameters of multiple-mode transducers from the electrical admittance. These data could be used as starting values for subsequent iterative refinement. In this paper, an approach is presented to determine the equivalent circuit of broadband transducers with a nearly flat TVR and 2 resonances in the impedance in the band of interest. The equivalent circuit has lumped fre- quency-independent components. Approximate equivalent circuit parameters are first estimated using 7 measured values. Then, they are refined using regression analysis. The method is illustrated by computing the conductance and susceptance of the equivalent circuits of 1) a molded, free-flooded, radially polarized, piezoelectric ceramic shell, 2) a Tonpilz wideband transducer, and 3) a piezocompos- ite transducer, in water and comparing them with the measured values. II. Equivalent Circuit An electrical equivalent circuit for an underwater pi- ezoceramic transducer with 2 resonances in the band of interest is shown in Fig. 1. Each of the 2 arms of the circuit with an R, an L, and a C corresponds to a reso- nance. The resistance R corresponds to the modal loss, the inductance L to the modal mass, and the capacitance C to the modal compliance. C0 is the clamped capacitance and is sufficient to model the transducer when there is no vibration due to either inertial clamping or clamped boundaries. At low frequencies, the capacitance is C0 + C1 + C2. The input electrical admittance Y of the network shown in Fig. 1 can be written as Y I V G jB j C R j L R j L j C j C = = + = + + +( ) + + +( )w w ww w0 1 1 1 1 2 2 1 2 1 1 , (1) Manuscript received April 17, 2007; accepted February 24, 2008. The authors are with the Naval Physical & Oceanographic Labora- tory, Cochin, India (e-mail: tsonpol@vsnl.com). Digital Object Identifier 10.1109/TUFFC.899 where G and B are the input electrical conductance and susceptance, respectively. The subscripts 1 and 2 corre- spond to the first and second arms in the equivalent circuit shown in Fig. 1. Eq. (1) is easily rearranged to obtain G R R L R R L C C = + -( ) + + -( ) 1 1 2 1 1 1 2 2 2 2 2 1 2 2 w ww w (2) and B j C L R L L R L C C C C = - -( ) + -( ) - -( ) + -( ) w w w w w w w w w 0 1 1 1 1 2 1 1 1 2 2 1 2 2 2 2 1 2 2 .. (3) The equivalent circuit for a transducer with only one reso- nance is obtained by removing L2, C2, and R2. Ebenezer and Ravi [9] considered a transducer in air with one resonance frequency in the band of interest. They presented a method to determine the values of the cir- cuit elements by using 6 critical points in the admittance curve, as described in Fig. 2. These are the measured val- ues of 1) the frequency fs, at which the conductance G is maximum; 2) the maximum value of G (Gmax); 3) the frequency fx, which is less than fs, at which G ≅ Gmax/2; 4) the value of G at fx (Gx); 5) the frequency fy, at which B is locally maximum; and 6) the maximum value of B, i.e., By = Bmax. They derived explicit expressions for the val- ues of the circuit elements in terms of the measured values and, therefore, made it very easy to use the method. They also showed that there is good agreement between the measured and calculated conductance and susceptance at all frequencies near resonance even though information at only spot frequencies is used. The method can be used to determine the approximate equivalent circuit of an under- water transducer. In this method [9], the circuit parameters are deter- mined from the 6 values of frequencies, G, and B, using the relations R G = 1 max , (4) L RR Gx s x x = -( ) - æ è ççç ö ø ÷÷÷ 2 2w w w , (5) C L s = 1 2w , (6) C B L R L y y y yC y yC 0 1 2 1 2 1= æ è ççç ö ø ÷÷÷÷ + -( ) + -( ) é ë ê ê ê ê ê ê ù û ú ú ú úw w w w w úú ú , (7) where ωs = 2πfs, ωx = 2πfx, and ωy = 2πfy. The values of the 7 circuit elements in Fig. 1 can be de- termined by extending the approach used in [9]. However, this will require the solution of 7 simultaneous equations. Therefore, an alternative method is used and illustrated by finding the equivalent circuits of 3 types of transduc- ers. 2080 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, vol. 55, no. 9, SEPTEMBER 2008 Fig. 1. Equivalent circuit of a piezoelectric ceramic transducer with 2 resonances in the band of interest. Fig. 2. Definition of various critical points in the admittance curve used for calculating the equivalent circuit parameters using (4)–(7). In this method, the circuit elements of the 2 arms cor- responding to 2 resonances are independently determined. First, the values of R1, L1, C1, and C0 shown in Fig. 1 are approximately determined from (4)–(7) by ignoring R2, L2, and C2 and using the measured values of fs, Gmax, fx, Gx, fy, and By, in water, in the neighborhood of the first resonance. Similarly, R2, L2, and C2 are approximately determined from (4)–(7) by ignoring R1, L1, and C1 and using the measured values of fs, Gmax, fx, Gx, fy, and By, in water, in the neighborhood of the second resonance. It is assumed that the 2 resonances do not interfere with each other and their circuit parameters are determined independently. However, this assumption is used only to obtain the approximate initial values for the regression analysis to follow subsequently and, therefore, is not of much consequence. III. Regression Analysis The values of the equivalent circuit parameters com- puted from (4)–(7) are used as approximate initial values, and a nonlinear regression analysis is done to refine them. Refinement is achieved by minimizing the sums of the squares of the differences between the measured conduc- tance and the conductance computed using (2) at 400 spot frequencies in the band of interest. The method used by Ramesh and Vishnubhatla [10] to determine the effective piezoelectric coefficients of a piezocomposite is used for regression analysis. In this method, a Taylor series expan- sion for the conductance of the equivalent circuit with the values of the circuit elements as variables and explicit ex- pressions for the first derivatives of the conductance with respect to these values are used. Let R1, L1, C1, R2, L2, and C2 be the approximate initial values of the equivalent circuit elements obtained in the previous section. They are to be refined. Let ΔR1, ΔL1, ΔC1, ΔR2, ΔL2, and ΔC2 be the errors (correc- tions) in these parameters to be determined at each itera- tion step. Expanding the electrical conductance G(ω) in a Taylor’s series in terms of the respective correction terms yields G G R L C R L G R R G L L G C C ( ) , , , , ,w w= ( ) +¶ ¶ + ¶ ¶ + ¶ ¶ 1 1 1 2 2 2 1 1 1 1 1 1 ,C + D D D ¶¶ ¶ + ¶ ¶ + ¶ ¶ +G R R G L L G C C 2 2 2 2 2 2D D D ... (8) The higher-order terms that contain second derivatives of G are not explicitly shown in (8) and are negligible if the approximate initial values are close to the correct values. The sum of squares of residuals is written as S G Gi i i N = - ( )( ) = å ( ) ,expw w 2 1 (9) where G( )w is given by (8), G iexp w( )are the measured conductance values, and N is the number of data points. To minimize S, the partial derivatives of S with respect to the correction terms are equated to zero. Therefore, ¶ ¶ = ¶ ¶ = ¶ ¶ = ¶ ¶ = ¶ ¶ S R S L S C S R S L ( ) , ( ) , ( ) , ( ) , ( D D D D D 1 1 1 2 0 0 0 0 22 2 0 0 ) , ( ) ,= ¶ ¶ = and S CD (10) where the partial derivatives of G with respect to the cir- cuit parameters are given by ¶ ¶ = -( ) - + -( ) G R L R R L j j C j j j j C j w w w w 1 2 2 2 1 2 , (11) ¶ ¶ = - -( ) + -( ) G L R L R L j j j C j j j C j 2 2 1 2 1 2 w w w , (12) ¶ ¶ = - æ è çççç ö ø ÷÷÷÷ + -( ) G C R R L j j C j L C j j j C j j2 1 2 3 2 2 1 2 w ww , (13) where j = 1,2. The correction terms ΔR1, ΔL1, ΔC1, ΔR2, ΔL2, and ΔC2 are obtained by solving (10) and are added to the initial values to get the new set of parameters. Accuracy is improved by taking these values as new set of initial values and repeating the process a few times. The process converges after a few iterations. It is terminated when the variations in the successive values of the coefficients are less than the preset cut-off factor (C), so that a k a k a k C ii i i ( ) ( ) ( ) , - - - £ = ¼1 1 1 6for (14) where a ki( )and a ki( )- 1 are the values of the coefficients at kth and (k − 1)th steps. The value of C0 is not refined, as it does not occur in the expression for G in (2). As expected, the agreement between the measured and the computed values of magnitude of admittance is better than the agreement between the measured and the com- puted values of conductance and susceptance, when the magnitude of admittance is used in the regression analy- sis, instead of the measured conductance. Therefore, the choice of function used in the regression analysis should be based on the subsequent use of the equivalent circuit. 2081RAMESH AND EBENEZER: equivalent circuit for broadband underwater transducers IV. Numerical Results Equivalent circuit parameters of 1) a molded, free- flooded, radially polarized, piezoelectric ceramic shell, 2) a Tonpilz wideband transducer, and 3) a piezocompos- ite transducer in water are determined using the present method and refined using regression analysis. The con- ductance and susceptance are computed using the refined values of the circuit parameters and compared with the measured values. The measured conductance and susceptance of a mold- ed, free-flooded, radially polarized, piezoelectric ceramic cylindrical shell are shown in Fig. 3 by dashed lines. The values of the circuit elements obtained using explicit ex- pressions that are input to the regression analysis and the refined values are shown in Table I. In this case, refinement causes less than 10% change in L1, C1, and R2, about 15% change in R1, and about 33% change in L2 and C2. G and B obtained using (2) and (3), respectively, and the initial values given in Table I are shown in Fig. 3 by solid lines. It is seen that the measured and calculated values are in reasonably good agreement in the entire band of interest. In Fig. 4, the measured conductance and susceptance are compared with the corresponding values computed using the refined values shown in Table I. It is seen that agree- ment is even better than in Fig. 3. Fig. 5 shows the conductance and susceptance of a wideband Tonpilz transducer. The dashed lines are the measured data and the solid lines are the data computed using the refined values of the equivalent circuit param- eters given in Table II. It can be seen from the figure that the agreement is very close. In addition, the variations in all of the circuit element values, after refinement, are less than 8%, as seen from Table II. Analysis of piezocomposite transducers that are lossy and have 2 resonances in the frequency range of interest shows that the conductance, computed using the refined values given in Table III, agrees very well with measured data. The agreement is slightly poor in the case of sus- 2082 IEEE TRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCY CONTROL, vol. 55, no. 9, SEPTEMBER 2008 Fig. 3. The measured admittance of a molded, free-flooded, radially po- larized piezoelectric ceramic cylindrical shell and the admittance com- puted using the equivalent circuit shown in Fig. 1 and the initial values given in Table I. TABLE I. Values of Equivalent Circuit Elements of a Molded, Free-Flooded, Radially Polarized Piezoelectric Ceramic Cylindrical Shell. Parameter Initial value Refined value Correction (%) C0 (nF) 15.25 — — R1 (kΩ) 2.157 2.468 +14 L1 (H) 0.1103 0.1150 +4 C1 (nF) 1.060 1.005 −5 R2 (kΩ) 1.570 1.670 +6 L2 (H) 0.0223 0.0158 −29 C2 (nF) 1.160 1.593 +37 Fig. 4. The measured admittance of a molded, free-flooded, radially po- larized piezoelectric ceramic cylindrical shell and the admittance com- puted using the equivalent circuit shown in Fig. 1 and the refined values given in Table I. Fig. 5. The measured admittance of a broadband Tonpilz transducer and the admittance computed using the equivalent circuit shown in Fig. 1 and the refined values given in Table II. ceptance, as shown in Fig. 6. The initial and the refined values of the circuit parameters are given in Table III. Refinement causes significant variations (of 15–33%) in all of the parameters. In general, it is observed that both G and B are in good agreement even though only conductance is used in the regression analysis. This indicates that the form of the equivalent circuit in Fig. 1 is appropriate. V. Conclusions A method is presented to determine the equivalent cir- cuit of broadband underwater piezoelectric ceramic trans- ducers. In this method, approximate values of the circuit elements are first obtained using explicit expressions. These approximate values are then used as initial values and iterated to minimize the mean square difference be- tween the measured conductance and the computed con- ductance of the equivalent circuit. It is found that there is good agreement between the measured and computed conductances and susceptances even though only the con- ductance is used in the iteration. This indicates that the phase of the admittance is also correctly represented. The method is illustrated using measured data for a free-flood- ed, radially polarized piezoelectric cylinder, a wideband Tonpilz projector, and a piezocomposite projector. The equivalent circuit obtained using this method can be used to analyze and design matching filters that will interface the projector with the power amplifier to ensure maxi- mum power transfer. Acknowledgments The authors thank Sreejith S. Pillai for carrying out measurements on a free-flooded cylinder, M. R. Subash Chandra Bose for providing us with a wideband Tonpilz transducer, and Director, Naval Physical and Oceano- graphic Laboratory (NPOL), Kochi, India, for encourage- ment, providing facilities, and permission to publish this paper. References [1] S. C. Butler, “Development of a high power broadband doubly res- onant transducer,” in Proc. Underwater Defense Technol. (UDT- 2001), Hawaii, CD ROM, PII-5, 2001. [2] J. M. Lee, H. S. Mok, G. H. Choe, T. M. Kim, D. Y. Kwon, and K. T. Han, “Design of power amplifier and matching network for so- nar deriving system,” in Proc. Underwater Defense Technol. (UDT- 2002), Korea, CD ROM, 8A–3, 2002. [3] D. Stansfield, Underwater Electroacoustic Transducers, Bath, UK: Bath University Press, 1990. [4] L. Shuyu, “Equivalent circuits and directivity patterns of air-cou- pled ultrasonic transducers,” J. Acoust. Soc. Am., vol. 109, no. 3, pp. 949–957, 2001. [5] W. J. Marshall and G. A. Brigham, “Determining equivalent circuit parameters for low figure of merit transducers,” Acoust. Res. Lett. Online, vol. 5, no. 3, pp. 106–110, 2004. [6] S. Sherrit, H. D. Wiederick, B. K. Mukherjee, and M. Sayer, “An accurate equivalent circuit for the unloaded piezoelectric vibrator in the thickness mode,” J. Phys. D Appl. Phys., vol. 30, pp. 2354–2363, 1997. [7] D. Church and D. Pincock, “Predicting the electrical equivalent of piezoceramic transducers for small acoustic transmitters,” IEEE Trans. Sonics Ultrason., vol. SU-32, no. 1, pp. 61–64, 1985. [8] R. Coates and P. T. Maguire, “Multiple-mode acoustic transducer calculations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 36, no. 4, pp. 471–473, 1989. [9] D. D. Ebenezer and N. Ravi, “An equivalent circuit approach to de- termine the in-situ radiation impedance of underwater electroacous- tic projectors,” J. Acoust. Soc. India, vol. 21, pp. 110–115, 1992. [10] R. Ramesh and R. M. R. Vishnubhatla, “Estimation of material pa- rameters of lossy 1-3 piezocomposite plates by non-linear regression analysis,” J. Sound Vibrat., vol. 226, no. 3, pp. 573–584, 1999. 2083RAMESH AND EBENEZER: equivalent circuit for broadband underwater transducers TABLE II. Values of Equivalent Circuit Elements of a Wideband Tonpilz Transducer. Parameter Initial value Refined value Correction (%) C0 (nF) 7.133 — — R1 (kΩ) 1.368 1.404 +2.7 L1 (H) 0.2855 0.2749 −3.7 C1 (nF) 2.259 2.350 +4.0 R2 (kΩ) 2.753 2.861 +3.9 L2 (H) 0.4029 0.4350 +8.0 C2 (nF) 3.082 2.873 −6.8 TABLE III. Values of Equivalent Circuit Elements of a