Electroacoustic analysis of an electret loudspeaker using combined finite-element and lumped-parameter models

Electroacoustic analysis of an electret loudspeaker using

combined finite-element and lumped-parameter models

Mingsian R. Bai,a兲 Rong-Liang Chen, and Chun-Jen Wang

Department of Mechanical Engineering, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsin-Chu 300, Taiwan

共Received 1 May 2008; revised 11 March 2009; accepted 12 March 2009兲

An unconventional type of electrostatic loudspeaker is presented in this paper. The loudspeaker made of thin, light, and flexible electret material lends itself well to the space-concerned applications. Electrical impedance measurement reveals that the coupling between the electrical system and the mechanical system is weak, which renders conventional parameter identification based on electrical impedance measurement impractical. A different approach is thus employed to model the electret loudspeaker. To predict the loudspeaker’s dynamic response, finite-element analysis共FEA兲 is conducted on the basis of a simple model and a full model. In the simple model, FEA is applied to model the electret membrane, leaving the rest of system as rigid parts. In the full model, FEA is applied to model the entire membrane-spacer-back plate assembly. Velocity response of the membrane subject to a uniformly distributed force is calculated using FEA harmonic analysis. Mechanical impedance is then calculated with the velocity response. The acoustical impedance due to the back cavity, pores, and the radiation loading at the front side is calculated by theoretical formulas. The volume velocity of the membrane and the resulting on-axis sound pressure level are predicted with electrical-mechanical-acoustical analogous circuits. The response data predicted by the simulation compare very well with experimental measurements.

© 2009 Acoustical Society of America. 关DOI: 10.1121/1.3117377兴

PACS number共s兲: 43.38.Bs, 43.38.Ja, 43.40.Dx 关AJZ兴 Pages: 3632–3640

I. INTRODUCTION

A flat type of loudspeaker based on electret technology is presented in this paper. The loudspeaker is made of thin, light, and flexible electret material, which lends itself very well to space-concerned applications as demanded by many 3C 共computer, communication, and consumer electronics兲 products. This paper aims to model the electret loudspeaker and assess the acoustical performance using a combined finite-element and lumped-parameter model. Being able to simulate the response of the transducer is crucial to optimiz-ing the performance of this special type of transducer.

The principle of the electret loudspeaker resembles that of electrostatic loudspeakers. An electrostatic loudspeaker exploits the varying electrostatic force generated between two charged plates separated by an air gap. In the 1930s, there were a number of practical electrostatic loudspeakers invented.1 McLachlan2 conducted theoretical investigations of sound power and pressure at resonant and non-resonant frequencies on a stretched-membrane electrostatic loud-speaker. On the other hand, electret is a dielectric material that has a quasi-permanent electric charge or dipole polariza-tion. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent mag-net. There is a similarity between electrets and the dielectric layer used in capacitors; the difference being that dielectrics in capacitors possess an induced polarization that is only transient, dependent on the potential applied on the

dielec-tric, while dielectrics with electret properties exhibit quasi-permanent charge storage or dipole polarization in addition. Paajanen et al.3modeled the electret film using a simplified model involving multiple parallel air gaps. Reciprocity of the sensor and actuator is demonstrated experimentally using the model. Mellow and Karkkainen4 calculated the free-space radiation of a tensioned circular membrane. Near-field and far-field pressure responses and efficiency were calculated based on the diaphragm impedance. The on-axis pressure response calculated using this method was compared with that obtained using a finite-element model. Medley et al.5 analyzed an electrostatic transducer analytically and numeri-cally. The frequency response of the transducer with layered construction was calculated and compared with experimental data.

Electret transducers have advantage over conventional electrostatic transducers to enable simple and compact elec-troacoustic construction without external polarization circuit and power supply. A celebrated example is the electret con-denser microphone invented at Bell laboratories in 1962 by Sessler and West,6using a thin metalized Teflon foil. Rather than the well established microphone technology, the recip-rocal application of electret material to loudspeakers is at-tempted in this paper. Over the past years, various polymer materials such as polyethylene, polystyrene, polyurethane, polyethyleneterephtalate, Teflon 共FEP and PTFE兲, polyvi-nylidene fluoride, and polypropylene共PP兲 have been used as electret materials.7,8 These are usually homogeneous solid materials. Paajanen et al.8suggested a new multipurpose ma-terial, electromechanical film 共EMFi兲. The EMFi is a thin, cellular, biaxially oriented PP film that can be used as an

a兲_{Author to whom correspondence should be addressed. Electronic mail:} msbai@mail.nctu.edu.tw

electret. Having a special voided internal structure and high resistivity, it is capable of storing large permanent charge. EMFi can be applied to sensors, switches, loudspeakers, spe-cial transducers, etc. Saarimaki et al.9 developed heat-resistive sensors and actuators using electromechanical cyclo-olefin-based polymer films. They described in detail the manufacturing of electromechanical and electrostatic transducers used in the audible frequency range. Recently, Chiang and Chen10 proposed a new flexible electret loud-speaker using nano-technology. The electret diaphragms are fabricated using fluoro-polymer with nano-meso-micro pores precharged by the corona method. Heydt et al.11 measured the acoustical response produced by electrostrictive polymer film loudspeakers. Measurements of harmonic distortion are also shown, accompanied with results demonstrating reduced harmonic distortion achieved using a square-root wave shap-ing technique.

This paper aims at establishing a simulation platform for electret loudspeakers with the main structure consisting of a membrane, a spacer grid, and a back plate. Due to the effect of the spacer grid, the membrane is rigidly constrained at the grid such that the motional impedance presented in the ter-minal electrical impedance is almost negligible. As we shall see later in the presentation, the characteristics of the me-chanical system are not present in the electrical domain and the effective electrical impedance resembles that of a capaci-tor. Electrical impedance measurement reveals that the cou-pling between the electrical system and the mechanical sys-tem is weak, which renders conventional parameter identification based on electrical impedance measure-ment12–14impractical in the case of electret loudspeakers. In this paper, a different approach is thus employed to model the electret loudspeaker by taking advantage of the fact that the electrical and the mechanical systems are weakly coupled. Specifically, the electrical response is treated as a capacitor without considering the mechanical loading, whereas the mechanical response is calculated by using the unloaded electrical response as the input to the mechanical systems. Finite-element analysis 共FEA兲 is employed in a simple model as well as a full model to predict the loud-speaker’s dynamic response. The FEA model is tuned by matching the resonance frequency predicted by a single-cell model with the velocity response measured by a laser vibro-meter. The mechanical impedance is estimated by the aver-age velocity response subjected to a uniformly distributed force. Furthermore, the acoustical impedance due to the back cavity, pores, and the radiation loading at the front side is calculated by the theoretical formulas. Using this hybrid model, the volume velocity of the membrane and the on-axis sound pressure level共SPL兲 can be calculated with accuracy.

II. OPERATING PRINCIPLES OF AN ELECTRET LOUDSPEAKER

A sample of electret loudspeaker with length 101 mm and width 41 mm is shown in Figs.1共a兲and1共b兲. The main structure of the loudspeaker includes a membrane 共nano-porous fluoro-polymer兲, a spacer grid, and a perforated back plate共stainless steel兲 with 24.15% perforation ratio. The gap between the membrane and the back plate is 100␮m. Figure

1共c兲is the top-view of the loudspeaker structure, where the arrangement of the spacer grid is shown. The time-varying attraction force between the back plate and the membrane gives rise to the motion of the membrane. The device per se is a capacitor with capacitance varying with the distance be-tween the membrane and the back plate, as shown in Fig.

2共a兲. The static attraction force共Coulomb force兲 between the membrane and the back plate can be calculated using the following formula: F0= ␧S 2 E02 d2. 共1兲

The dielectric constant␧ is given by ␧ =C0d

S , 共2兲

where S is the area of action, E₀ is the effective dc bias voltage due to the precharged electret, d is the distance

be-membrane

electrode electrode

electrode back plate electrode

(b) (a) 101m m 41m m back plate spacer 1m m 8m m 4m m 8m m (c)

FIG. 1. The electret loudspeaker.共a兲 Photo 共front view兲. 共b兲 Photo 共rear view兲. 共c兲 Schematic showing the perforated back plate and the spacer grid.

tween the membrane and the back plate, and C₀is the static capacitance. Assume that the membrane motion is small and the gap between the membrane and the back plate is nearly constant. The dynamic attraction force produced by the driv-ing voltage can be approximated as

f =␧SE0e

d2 , 共3兲

where e is the driving voltage. The electret loudspeaker can be modeled with the electroacoustic analogous circuit in Fig.

2共b兲. The linearized ac dynamic equation can be expressed as follows: e = ZEi + 1 j␻CEM uM, 共4兲 f = 1 j␻CEM i + ZMuM, 共5兲 where ZE= 1 j␻C₀, 共6兲 CEM= ␧S q0 , 共7兲

i is the current, uM is the membrane velocity, ZM is the

共open-circuit兲 mechanical impedance of the diaphragm, ZEis

the 共blocked兲 electrical impedance, q₀ is the static charge, and C₀ is the static capacitance. Equations 共4兲 and共5兲 can also be expressed in matrix form

冋

册

冤

冥

冋

册

The fact that the impedance matrix is symmetric indicates that the device is a reciprocal transducer 共XDCR兲 with an associated transduction factor ␾

␾= C0E0

d =

C0 CEM

. 共9兲

Equations共4兲 and共5兲 can be rewritten as

e = 1 j␻C0 ic, 共10兲 f = ␾ j␻C0 ic+ ZmsuM, 共11兲 where ic= i +␾uM, 共12兲 Zms= ZM− 1 j␻C0/␾2 . 共13兲

Zmsis the short-circuit mechanical impedance, which is the

sum of the open-circuit mechanical impedance ZM and a

negative capacitor 共−C0/␾2兲. The negative capacitor can be removed if controlled sources are used in lieu of the trans-former coupling.

Figure 2共c兲shows the circuit with the mechanical sys-tem reflected to the electrical domain. Zme is the motional

impedance reflected to the electrical domain from mechani-cal system.

Zme=

Zms

␾2. 共14兲

Since the transduction factor共␾兲 is very small, the mechani-cal impedance reflected to the electrimechani-cal domain effectively becomes an open circuit. This is why the mechanical char-acteristics would not appear in the electrical impedance. In the following, we shall verify this point by using a practical measurement and FEA.

The terminal electrical impedance is measured from 400 Hz to 10 kHz using a 1.5 Vrms sweep-sine input. Figure

3共a兲 shows the experimental arrangement for the measure-ment, with symbols defined in the figure

+

_

e

1:

φ

C

+

-Z

+

-f

+

e

FIG. 2. The electret loudspeaker. 共a兲 Cross-section. 共b兲 Electroacoustic analogous circuit.共c兲 Analogous circuit reflected to the electrical domain.

Ze=

egG − er

er

R, 共15兲

where G is the gain of the voltage amplifier and Ze is the

terminal electrical impedance. The measured electrical im-pedance is shown in Fig. 3共b兲. It can be observed from the figure that the measured electrical impedance fits very well with the impedance of an ideal capacitor because the elec-tromechanical coupling is weak enough to conceal the mo-tional impedance.

兩Ze兩 =

1

␻C₀. 共16兲

Although the mechanical parameters cannot be extracted from the measured electrical impedance, the static capaci-tance共C0兲 can still be estimated as follows:

log C0= − log␻− log兩Ze兩. 共17兲

Using this formula, the static capacitance 共C0兲 can be esti-mated from the impedance log-log plot to be 7.9577 ⫻10−10_{F. For the electret loudspeaker sample under test, the} effective dc bias voltage 共E0兲 is approximately 600 V, the gap between the membrane and back plate is 0.1 mm, and the applied voltage共e兲 is 20 Vrms. Thus, the dielectric con-stant 共␧兲 is estimated by Eq. 共2兲 to be 1.9217⫻10−11_{. The} transduction factor 共␾兲 calculated using Eq. 共9兲 is 0.0048. The static and dynamic attraction forces estimated using Eqs.

共1兲 and共3兲 are 1.4324 N and 0.0955 Nrms, respectively.

III. THE MODEL OF THE ELECTRET LOUDSPEAKER

As mentioned previously, the electrical impedance of the electret loudspeaker is predominantly capacitive without be-ing influenced by the motional impedance, meanbe-ing the elec-tromechanical coupling is extremely weak. Therefore, the parameter identification based on electrical impedance mea-surement is not applicable to the electret loudspeaker. An alternative approach for modeling the electret loudspeaker will be presented next.

A single-cell model, a simple model, and a full model based on FEA are developed to model the electret loud-speaker. Specifically, FEA is applied to a single-cell on the membrane in the single-cell model, and FEA is applied to the membrane in the simple model, leaving the rest of system as rigid parts, whereas FEA is applied to the entire membrane-spacer-back plate assembly in the full model. The FEA is carried out with the aid of ANSYS®.15 In the single-cell model, the natural frequencies and the mode shapes of the membrane are calculated by the FEA modal analysis. This FEA result combined with the laser vibrometer measurement determines the material constants of the membrane. To evaluate the mechanical impedance of the membrane, the displacement frequency response of the membrane is calcu-lated and converted into velocity by using the FEA harmonic analysis. The ratio of the external force to the mean velocity of diaphragm in mechanical system is defined as the open-circuit mechanical impedance ZM. In order to facilitate the

integration of the mechanical system, the open-circuit me-chanical impedance 共ZM兲 defined in the following

lumped-parameter two-port formalism is calculated using the com-plex velocity response.

ZM=

F u

¯, 共18兲

where F is an external force applied in the finite-element model and u¯ denotes the mean of the complex velocity, which is obtained by the FEA harmonic analysis. Moreover, the impedance Zmethat is reflected from the mechanical

do-main to the electrical dodo-main can be calculated by Eq.共14兲. Figure4compares兩Zme兩 and the electrical impedance 兩ZE兩. It

can be observed from the figure that 兩Zme兩 is much greater

than 兩ZE兩. Therefore, the mechanical impedance reflected to

the electrical domain effectively becomes an open circuit. The mechanical characteristics would not appear in the elec-trical impedance. This justifies the weak coupling assertion between the electrical and the mechanical systems.

The FEA simulation is based on both the simple and the full models. The mechanical impedances are calculated using harmonic analysis of FEA. The mean velocity uM can be

obtained by solving the electrical-mechanical-acoustical analogous circuits of Fig. 2共b兲. From a membrane with ef-fective area SD, the on-axis pressure at a far field distance r

can be calculated using the simple source model p共r兲 = j␳0␻UM

2␲ e−jkr

r , 共19兲

where UM= SDuMis the volume velocity,␳0is the density of air共␳₀= 1.21 kg/m3_{兲, and k is the wave number.}

(a) (b) ch 2 eg er electret loudspeaker signal generator ch 1 _R analyzer resistor

 output voltage voltage cross resistor amplifier

FIG. 3. 共Color online兲 The electrical impedance measured at the terminals of the electret loudspeaker.共a兲 Experimental arrangement. 共b兲 The frequency response of the measured electrical impedance.

The FEA simulation is based on both simple and full models.

A. The single-cell model and the simple model

In the simple model, it is assumed that the back plate and the spacer are rigid, while the FEA models only the flexible membrane of the electret loudspeaker. The

mem-brane is modeled with the element “shell 41,” which is a three-dimensional element allowing membrane 共in-plane兲 stiffness but no bending共out-of-plane兲 stiffness. The element has four nodes and three degrees of freedom共Ux, Uy, and Uz兲

at each node.

We begin with tuning the single-cell FEA model with reference to the laser velocity response measurement. The mesh used in the finite-element model of a single-cell is shown in Fig.5. The boundary conditions are selected such that all degrees of freedom for the outer edge of the single-cell and the displacements at the spacer are zero. Using this approach, the determined material constants of the mem-brane are summarized in TableI.

Next, the simple FEA model and the associated mesh of the membrane are shown in Fig.5. The boundary conditions are selected such that all degrees of freedom for the outer edge of the membrane and the displacements at the spacer are set to be zero. The complex velocity and mechanical impedance are calculated using the FEA harmonic analysis.

101 102 103 104 104 105 106 107 108 109 Frequency (Hz) Im ped a nc e (ohm ) 123 123 me Z E Z

FIG. 4. The comparison between the frequency response functions of Zme and ZE.

edge

spacer

membrane

single cell

model

simple model

FIG. 5. 共Color online兲 The mesh plot of the simple FEA model and single-cell FEA model of the membrane. TABLE I. Material properties of the electret loudspeaker.

Parameters Spacer and back plate Membrane Young’s modulus共N/m2_{兲 201⫻10}9 _82⫻109

Poisson’s ratio 0.28 0.3

Density共kg/m3_{兲 8000 700}

B. The full FEA model

In the full model, FEA is applied to the whole structure of the electret loudspeaker consisting of the membrane, the spacer grid, and the back plate.16 The material constants as-sumed in the FEA are those obtained in the single-cell model 共TableI兲. The membrane, the spacer, and the back plate are

modeled using “shell 41,” “solid 45,” and “shell 63,” respec-tively. The solid 45 element has eight nodes and three de-grees of freedom共Ux, Uy, and Uz兲 at each node. The shell 63

element has four nodes and six degrees of freedom,共Ux, Uy,

Uz, ROTx, ROTy, and ROTz兲 at each node. The finite-element

mesh is shown in Fig. 6. The boundary conditions are se-lected such that all degrees of freedom for the outer edge of the electret loudspeaker are zero. The attraction forces be-tween the back plate and membrane are calculated using Eqs.

共3兲and共4兲. The effective mechanical impedance of the loud-speaker can also be established via the FEA harmonic analy-sis, as that of the simple model.

C. Modeling of the acoustical enclosure

Apart from the electrical and the mechanical systems, the modeling of the electret loudspeaker takes into account the acoustical loading resulting from the perforation of the back plate and the cavity between the back plate and the membrane. The back plate has 1000 square holes uniformly distributed over the metal sheet共perforation ratio=24.15%兲. The acoustical impedance of the perforation can be written as14 ZAP= RAP+ j␻MAP, 共20兲 RAP= ␳0 N␲ah2

冑

冋

冉

冊

册

冋

冉

冊

册

where N is the number of holes of the back plate, ␮is the kinematic coefficient of viscosity 共␮= 1.56⫻10−5_m2_{/s兲, a}

h

is the equivalent radius of the hole, b is the spacing between the center of the adjacent holes, and tpis the thickness of the

back plate. The cavity can be modeled as acoustic compli-ance

CA=

V

␳0c2

, 共23兲

where V is the volume of cavity and c is the speed of sound 共c⬇345 m/s at the room temperature兲. The analogous cir-cuit of the acoustical system of the electret loudspeaker is shown in Fig. 2共b兲. In particular, the circuit parameters to approximate radiation impedance are given by14

MA1= 8␳0 3␲2ah , 共24兲 R_A1=0.4410␳0c ␲ah 2 , 共25兲 RA2= ␳0c ␲ah2 , 共26兲 CA1= 5.94ah 3 ␳0c2 . 共27兲

The FEA-based simulation procedure for the electret loudspeaker can be summarized as follows.

共1兲 Obtain the resonance frequencies and velocity response from the single-cell FEA model of membrane.

共2兲 Measure the membrane velocity by a laser vibrometer and compare the data with the FEA results. Tune the FEA model with the experimental data.

共3兲 Using the simple FEA model, obtain the complex veloc-ity response of the membrane via FEA harmonic analy-sis.

共4兲 Using the full FEA model, obtain the complex velocity response of the membrane via FEA harmonic analysis. 共5兲 Calculate the mechanical impedance 共ZM兲 based on the

complex velocity response.

共6兲 Calculate the acoustical impedance 共ZA兲 due to the

per-forated back plate and the cavity by the theoretical for-mulas and combine the mechanical impedance and the acoustical impedance to form a coupled lumped-parameter model.

共7兲 Calculate the volume velocity using the coupled model. Calculate the on-axis pressure based on the volume ve-locity response.

共8兲 Compare the measured on-axis pressure with the re-sponse simulated using the simple and full FEA models.

IV. NUMERICAL AND EXPERIMENTAL INVESTIGATIONS

Experimental investigation was undertaken to validate the proposed simulation models of the electret loudspeaker. The experimental arrangement for an electret loudspeaker sample is shown in Fig. 7. The electret loudspeaker is mounted on a baffle of length 1650 mm and width 1350 mm according to the standard AES2-1984 共R2003兲.17 On-axis SPL and total harmonic distortion共THD兲 of the electret loud-speaker are measured using this setup. A 20 Vrms swept-sine signal is used to drive the electret loudspeaker in the fre-membrane

spacer back plate

quency range 20 Hz– 20 kHz. The on-axis SPL is measured by using a microphone positioned at 10 cm away from the loudspeaker.

A. Comparison of the measured and the simulated responses

Simulation of the membrane response was carried out using the single-cell finite-element model. The fundamental resonance frequency predicted by the FEA modal analysis is 849.43 Hz, with the associated mode shape shown in Fig.

8共a兲. Using the FEM harmonic analysis, complex displace-ment is calculated and converted into the velocity. The fun-damental resonance frequency of the velocity response mea-sured using the laser vibrometer is 818.75 Hz, which is in close agreement with the foregoing FEA result. Figure8共b兲 compares the velocity responses obtained from the experi-ment and simulation. Except for some minor discrepancies at high frequencies, the simulated responses共dotted lines兲 com-pare well with those of the measured results 共solid lines兲. Some peaks of the measured frequency response may corre-spond to unmodeled high-order modes of the membrane. Those modes cannot be predicted by the single-cell model.

On the basis of the material constants determined in the single-cell model, simulation can be conducted for the elec-tret loudspeaker using the simple and the full models. FEA modal analysis reveals that the fundamental resonance fre-quency of the simple model is 811.6 Hz, with the associated mode shape shown in Fig.9共a兲. In the fundamental mode, a drumming motion appears locally in phase at every cell of the membrane. The fundamental resonance frequency found in the full model is 789.3 Hz, with the associated mode shape shown in Fig. 9共b兲. Clearly visible is that the entire membrane-spacer-back plate assembly moves in phase col-lectively in this fundamental mode.

Next, the mechanical impedance of the membrane is cal-culated with the aid of the FEA harmonic analysis. Combin-ing the mechanical impedance and the acoustical impedances due to the cavity and radiation enables the estimation of the volume velocity produced by the electret loudspeaker, with which the on-axis SPL can be calculated. Figure 10 com-pares the on-axis SPL obtained from the simulation and the experiment, respectively. The FEA simulation is based on both the simple and the full models. It can be observed that in lower frequencies the on-axis SPL response predicted by

the FEA simple model is in good agreement with the mea-surement. The discrepancy between the measured low-frequency response and the FEA full model could be due to the effect of the spacer and the back plate, which may induce some unmodeled low-frequency motions. In high frequen-cies, however, the simple model fails to capture the response due to the flexural modes of the structure. By contrast, the FEA full model seems to have predicted the high-frequency response better than the simple model. The on-axis SPL re-sponse calculated using the full FEA model matches reason-ably well the measured response.

B. Nonlinear distortion

In order to assess the nonlinear distortion of the electret loudspeaker, THD is calculated from the measured frequency response of the on-axis SPL.18

THD =

冑

where pnf is the sound pressure magnitude of the nth

har-monic in the spectrum and p_1f is the sound pressure magni-electret loudspeaker

microphone

signal generator

analyzer

FIG. 7. 共Color online兲 Experimental arrangement of the on-axis SPL mea-surement for the electret loudspeaker.

(a)

(b)

FIG. 8. 共Color online兲 The simulation results obtained using the single-cell FEA model.共a兲 The fundamental mode shape of membrane. 共b兲 Comparison of the simulated velocity response with the measured response.

tude of the input fundamental frequency. The measured THD of the electret loudspeaker sample is shown in Fig.11. It can be observed that the average THD is about 17% and the THD is greater than 10% in the frequency range above the funda-mental resonance frequency. The maximum THD even reaches about 20% in the range 2 – 3 kHz. It may be caused by the structural resonance of the loudspeaker. Informal lis-tening tests revealed that ringing artifact is audible for some music signals involving percussion instruments.

V. CONCLUSIONS

A simulation technique for an electret loudspeaker based on the FEA alongside electroacoustical modeling has been

presented in this paper. As an early attempt for the transducer of this kind, the loudspeaker element has an extremely low sensitivity. The coupling from the mechanical to the electri-cal circuit was simply too low to make parameter measure-ments based on electrical impedance measuremeasure-ments. Due to the weak coupling between the electrical and the mechanical systems, the electrical impedance-based parameter identifica-tion procedure is not applicable to the electret loudspeaker. An alternative approach has been presented in the paper to model the electret loudspeaker. FEA was exploited for mod-eling the loudspeaker with a simple model and a full model. The mechanical impedance, the volume velocity of mem-brane, and the on-axis pressure can be predicted using the coupled electroacoustical model.

ACKNOWLEDGMENTS

The work was supported the National Science Council in Taiwan, under Project No. NSC 95-2221-E-009-009-MY2. Special thanks also go to the Electronics and Optoelectronics Research Laboratories, Industrial Technology Research Insti-tute in Taiwan.

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