An Analytical Model for Bandwidth Enhancement of Air ...downloads.hindawi.com/journals/js/2019/3896965.pdf · Capacitive micromachined ultrasonic transducers (CMUTs) ... tional piezoelectric

Research ArticleAn Analytical Model for Bandwidth Enhancement ofAir-Coupled Unsealed Helmholtz Structural CMUTs

Xin Liu 12 Yuanyu Yu123 JiujiangWang12 Sio Hang Pun 1

Mang I Vai12 and Peng UnMak 24

1 State Key Laboratory of Analog and Mixed-Signal VLSI University of Macau 999078 Macau2Department of Electrical and Computer Engineering Faculty of Science and Technology University of Macau 999078 Macau3School of Information Engineering Lingnan Normal University Zhanjiang 524048 China4On Leave to Acoustic Wave Physics Group Department of Chemical Engineering and Biotechnology University of CambridgeCambridge CB3 0AS UK

Correspondence should be addressed to Peng UnMak fstpumumedumo

Received 27 December 2018 Revised 6 March 2019 Accepted 27 March 2019 Published 17 April 2019

Academic Editor Chengkuo Lee

Copyright copy 2019 Xin Liu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Capacitive micromachined ultrasonic transducers (CMUTs) were reported to own high potential in air-coupled ultrasonicapplications such as noncontact nondestructive examination and gas flow measurement The unsealed CMUTs which utilizedthe squeeze film effect were reported to overcome the narrow output pressure bandwidth of the conventional sealed CMUTsin air operation This kind of unsealed CMUTs can also be regarded as Helmholtz resonators In this work we present the air-coupled unsealed Helmholtz structural CMUTs which utilize both the squeeze film effect and the Helmholtz resonant effect toenhance the output pressure bandwidth Based on the mechanism of vibration coupling between membrane and air pistons inmembrane holes we propose an analytical model to aid the design process of this kind of CMUTs We also use finite elementmethod (FEM) to investigate this kind of CMUTs for our analytical model validation The FEM results show that the significantbandwidth enhancement can be achieved when the Helmholtz resonant frequency is designed close to the fundamental resonantfrequency of the CMUTmembrane Compared with the conventional sealed CMUT cell the 4-hole unsealed Helmholtz structuralCMUT cell improves both the 3-dB fractional bandwidth and SPL-bandwidth product around 35 times Furthermore it is foundthat with more holes under the same hole area ratio or with a smaller ratio of the cavity height to the viscous boundary layerthickness the Helmholtz resonant effect becomes weaker and thus the output pressure bandwidth decreases

1 Introduction

Capacitive micromachined ultrasonic transducers (CMUTs)were invented as an alternative to conventional piezoelec-tric ultrasonic transducers in early 1990s Compared withthe piezoelectric ultrasonic transducers CMUTs have manyadvantages such as they are suitable for batch production andcan be easily integrated with modern electronics Conven-tional piezoelectric ultrasonic transducers typically requirematching layers to compensate the acoustic impedancemismatch between piezoelectric materials and surroundinggaseous or liquid media CMUTs feature a better impedancematching with gaseous and liquid media and therefore theycan transmitreceive ultrasound more efficiently into gas andliquid without the matching layer [1]

In many ultrasonic applications the transducers wereused in immersion [2 3] However in some applicationssome objects cannot contact with liquid couplants such aspaper products and electronic products [4] Also there aremany other air-borne ultrasonic applications such as air-coupled ultrasonic imaging [4] ultrasonic distance measure-ment [5] anthropometrical measurement of scoliosis [6]and gas flow measurement [7] CMUTs ideally are suitedfor air-coupled applications because of their good acousticimpedance matching with air

For air-coupled ultrasonic applications transducers withwide bandwidth have sharp ultrasonic pulses and shortdecay time of the vibration amplitude thus providing highresolution and a small ldquodead zonerdquo where objects cannot bedetected [8] Wide bandwidth air-coupled transducers are

HindawiJournal of SensorsVolume 2019 Article ID 3896965 16 pageshttpsdoiorg10115520193896965

2 Journal of Sensors

(a)

Vdc

Vac

membrane

substrate

R

t

h

top electrode2a

cavityhole holeinsulator layer

bottom electrode

support supportair pistons

d

Ｌ1

ＮＣＨＭ

(b)

Figure 1 (a) A 4-hole unsealed circular CMUT cell with Helmholtz resonator structure (b) Cross-sectional diagram of this cell along dashedline of (a)

needed for applications such as echo profile evaluation anddistance measurement with high resolution [8] However theconventional vacuum sealed air-coupled CMUTs still own anarrow bandwidth For instance for the CMUT array withonly one cell radius the 6-dB fractional bandwidth is about1 [9] To overcome this unsealed CMUTs were proposedto achieve wider bandwidth by utilizing the squeeze filmeffects [10ndash13] For unsealed CMUTs the vented cavity notonly introduces the squeeze film effects but also constructs aHelmholtz resonator with the vented holes The Helmholtzresonator comprised a cavity with an opening denoted asldquoneckrdquo When an excitation with a specific frequency rangeis applied a volume of air in the neck would vibrate due tothe elasticity of the air inside the cavity and thus radiatessound waves into surroundings This is called Helmholtzresonant effect [14] The Helmholtz resonant effect can alsoexert an influence on the device performance this phe-nomenon was reported but there was a lack of analyticalexplanation in [11] In [15] the Helmholtz resonant effectwas used to enlarge the bandwidth of the CMUTrsquos out-put but in its FEM simulation the squeeze film dampingwas neglected Actually with proper adjustment of volumeratio between vented hole and cavity the performance ofthe device would be benefited by the vibration couplingbetween the air pistons vibration system of Helmholtzresonator and the membrane vibration system of CMUTcell This vibration coupling was widely used to enhancethe low frequency response of musical instrument such asacoustic guitar [16] and vented-box loudspeaker systems[17]

In this paper we present an air-coupled unsealed CMUTcell with Helmholtz resonant cavity which utilizes both thesqueeze film effect and Helmholtz resonant effect to enhancethe output pressure bandwidth of the device For conve-nience we hereafter call it as Helmholtz structural CMUTWe propose an analytical model to explain the vibrationcoupling working mechanism of this kind of CMUT and aidthe design process of this kind of CMUT In order to validatethis analytical model and provide deeper investigation on thiskind of Helmholtz structural CMUT the FEM model is builtwith COMSOL Multiphysics version 52a (COMSOL Inc

Stockholm Sweden) to simulate three cases of 4-hole CMUTswith different hole radii

2 Methodologies

Figure 1(a) shows an air-coupled unsealed Helmholtz struc-tural CMUT cell with 4 holes in its membrane the cross-sectional illustration of this CMUT cell along the red dashedline in (a) is shown in Figure 1(b) Major components ofthis CMUT cell are a circular membrane with radius 119877 andthickness t a cavity with height h holes with radius a thethin metal layer as full top electrode and one insulator layerwith thickness tins to avoid short circuit between electrodesThe bottom electrode is located between the insulator layerand the substrate The rim of the membrane is clampedby support Using Helmholtz resonant effect and squeezefilm effect to enhance bandwidth several circular holes needto be punched through the membrane These holes andthe cavity form a Helmholtz resonator If a DC voltage isapplied between top and bottom electrodes the membranewill be activated by the electrostatic force induced by thevoltage and the restoring spring force due to the membranersquosstiffness When these two forces are equal the membrane willreach an equilibrium position Then when a small amplitudeAC voltage is superimposed the unbalance of these twoforces will be caused by the small perturbation of theelectrostatic force which leads to the membranersquos vibrationThe membranersquos vibration will drive the air pistons in theholes to vibrate due to the elastic property of the air insidethe cavity The vibration of the air pistons will also inducea pressure change in the cavity which leads to a force actedon the membrane Therefore the membranersquos vibration andthe air pistonsrsquo vibration interact with each other During theoperation of this air-coupled unsealed Helmholtz structuralCMUT cell the ultrasonic wave will be generated by bothvibrations of the membrane and the air pistons

21 Lumped-Parameter Analytical Model As Figure 2 showsinspired by the simple model of acoustic guitar [16] theHelmholtz structural CMUT cell can be simplified as aHelmholtz resonatorwith amovable top plateTheHelmholtz

Journal of Sensors 3

V

SA

+

-

F+

-

+

-

lma

xa

mp

kp

xp

Figure 2 Simplified model of Helmholtz structural CMUT cell

resonator is formed from the cavity of CMUT cell withvolume 119881 and an equivalent air piston with mass 119898119886 andtop surface area S where the top surface area 119878 is the sum oftop surface areas of all small holes And we can simplify themembrane as a movable top plate with mass 119898119901 top surfacearea 119860 and stiffness 119896119901

Then referencing from [16] we can develop a lumped-parameter analytical model of the air-coupled unsealedHelmholtz structural CMUT cell as follows

When the DC and AC bias voltages are superimposedthe top plate vibrates as a harmonic oscillator and drives theequivalent air piston to vibrate due to the elastic propertyof air cushion in the cavity We assume that the deflectedshape of the top plate can be neglected and the uniformdisplacement of the top plate is denoted as 119909119901 The uniformdisplacement of the equivalent air piston is taken as 119909119886 Thenwe take outward movement as positive the volume change ofthe cavity is Δ119881 = 119860119909119901 + 119878119909119886 (1)

Thus the pressure change as adiabatic compression in cavityis Δ119901 = minus120583Δ119881 (2)

where 120583=c2120588V c is sound velocity in air and 120588 is the densityof air The forces acted on plate and equivalent air piston dueto the pressure change can be represented by AΔp and SΔpand the equations of the vibrations of the top plate and theequivalent air piston can be written as

11989811990111988921199091199011198891199052 = 119865 minus 119877119901 119889119909119901119889119905 minus (119896119901 + 1205831198602) 119909119901 minus 120583119878119860119909119886 (3a)

11989811988611988921199091198861198891199052 = minus119877119886 119889119909119886119889119905 minus 1205831198782119909119886 minus 120583119878119860119909119901 (3b)

where 119865 is the driving force of top plate 119877119886 and 119877119901 areresistances to movement of the top plate and the equivalentair piston respectively From (3a) and (3b) we can findthat the top platersquos vibration and the equivalent air pistonrsquosvibration are coupled because they both depend on 119909119886 and119909119901 Namely the vibration of one of them can have effect onthe other

UncoupledHelmholtz resonator and uncoupled top platefirstly if we set S=0 in (3a) and 119909119901=0 in (3b) we can get theequations which describe the vibrations of the uncoupled topplate and uncoupled Helmholtz resonator respectively

11989811990111988921199091199011198891199052 = 119865 minus 119877119901119889119909119901119889119905 minus (119896119901 + 1205831198602) 119909119901 (4a)

11989811988611988921199091198861198891199052 = minus119877119886 119889119909119886119889119905 minus 1205831198782119909119886 (4b)

From (4a) we can get the resonant frequency of the uncou-pled top plate as

120596119901 = radic (119896119901 + 1205831198602)119898119901 (5)

Referencing from [18] this frequency can be calculatedmore accurately as the fundamental resonant frequency of aclamped circular membrane as

120596119901 = 2120587119891119901 = 2120587 sdot 047 1199051198772radic 119884[120588119898 (1 minus 1205902)] (6)

where 120588119898 120590 and119884 aremembranematerialrsquos density Poissonrsquosratio and Youngrsquos modulus respectively From (4b) theresonant frequency of an uncoupledHelmholtz resonator canbe solved as

120596ℎ = 2120587119891ℎ = radic 1205831198782119898119886 (7)

We substitute 120583=c2120588V 119898119886=Sl120588 into (7) and we can get

120596ℎ = 119888radic 119878119881119897 (8)

According to [19] the length of the Helmholtz resonatorrsquosneck 119897 should take an effective value leff=l+120575 for betterestimation thus

120596ℎ = 119888radic 119878[119881 (119897 + 120575)] (9)

where 120575 is the end correction Referencing from [19] the endcorrection 120575 should take the sum of the interior correction 120575119894and the exterior correction 120575119890 as follows

120575 = 120575119894 + 120575119890 = 048radic119878 (1 minus 125radic119899ℎ119900119897119890119886119877 ) + 048radic119878 (10)

where nhole is the number of holes in the membrane of theCMUT cell

Coupled Helmholtz resonator and top plate now wefocus on the coupled vibration system assuming in harmonicoscillation state and we solve (3a) and (3b) in frequencydomain by writing in phasor format 119909119901 = 119883119901119890119894120596119905 119909119886 =119883119886119890119894120596119905 119865 = 1198650119890119894120596119905 the amplitude of top platersquos velocity 119880119901 =119894120596119883119901 and the amplitude of equivalent air pistonrsquos velocity119880119886 = 119894120596119883119886 Similar to [16] we can get

119880119901 = 1198941205961198650 [(120596ℎ2 minus 1205962) + 119894120596120574119886]119898119901119863 (11a)

119880119886 = minus1198941205961198650119860120596ℎ2119898119901119878119863 (11b)


where119863 = (1205961199012 minus 1205962 + 119894120596120574119901) (120596ℎ2 minus 1205962 + 119894120596120574119886) minus 120596119901ℎ4 (12)

120574119886 = 119877119886119898119886 120574119901 = 119877119901119898119901

120596119901ℎ4 = 120596ℎ212059611988621205961198862 = 1205831198602119898119901

(13)

120596119901 and120596ℎ are the resonant frequencies of uncoupled top plateand uncoupled Helmholtz resonator respectively

From (11a) and (11b) the coupled vibration system shownin Figure 2 resonates while the moduli of 119880119886 and 119880119901 arrivemaximum (D=0) If we neglect the damping terms of i120596120574aand i120596120574p in (12) then this coupled vibration systemrsquos resonantfrequencies can be solved as

1205962plusmn = 1205961199012 + 120596ℎ22 plusmn radic(1205961199012 minus 120596ℎ2)2 + 4120596119901ℎ42 (14)

As shown in (6) and (9) for the uncoupled top plate andthe uncoupled Helmholtz resonator each of them only hasone resonant frequency From (14) we can see that unlike theuncoupled situation both the top plate and the equivalent airpiston have two resonant frequencies in the coupled vibrationsystem

For an air-coupled unsealed Helmholtz structural CMUTcell when calculating 120596ℎ by (9) the length of the neck 119897is equal to the membranersquos thickness 119905 And the lumpedparameters are calculated as follows119878 = 119899ℎ1199001198971198901205871198862 (15)

119860 = 1205871198772 minus 119878 (16)

119881 = 1205871198772ℎ (17)

119898119901 = 18119860119905120588119898 + 119898119903119901 (18)

119898119886 = 119878119905120588 + 119898119903119886 (19)where the factor 18 in (18) is from [20] 119898119903119901 is the acousticradiation mass of the plate and 119898119903119886 in (19) is the acousticradiation mass of the equivalent air piston Referenced from[21 22] the CMUT cell should be considered as a clampedplate source located on an infinite baffle and the acousticradiation mass of the plate is

119898119903119901 = minus120588119888119860120596 [[201198652 (2 (120596119888) radic119860120587)

((120596119888) radic119860120587)9 ]] 1198652 (119909) = minus (1199094 minus 911199092 + 504) 1198671 (119909)

minus 14119909 (1199092 minus 18) 1198670 (119909) + 14119909415120587 minus 1681199092120587(20)

where 119867119899(x) is the 119899th order Struve function The equivalentair piston is considered as a baffled piston source theradiation mass of the equivalent air piston is [14]

119898119903119886 = 119899ℎ119900119897119890120588119888 (1205871198862)120596 [21198671 (2 (120596119888) 119886)(2 (120596119888) 119886) ] (21)

In the simplification in Figure 2 the driving force 119865 is a har-monic force hence we take 119865 as the harmonic electrostaticforce according to [1]

1198650 = 10038161003816100381610038161198651198861198881003816100381610038161003816 = 1205760119860119881119889119888119881119886119888ℎ2 (22)

where 119881119889119888 119881119886119888 are DC voltage and AC voltage applied to thecell and 1205760 is the dielectric permittivity of air

The squeeze film effect includes squeeze film stiffeningeffect and squeeze film damping effect The proportion ofthe stiffening effect and damping effect is determined by thesqueeze number [20]

120590119904119902119890119890119911119890 = 121205831198861198941199031205961198772119875119886ℎ2 (23)

where 120583air is the dynamic viscosity of air 119875119886 is the ambientpressure and 120596 is the platersquos vibrating angular frequency Forthe air-coupled unsealed CMUT with perforated membranethe squeeze damping effect dominates the squeeze filmeffect because of the small squeeze number and the squeezestiffening effect can be neglected [20 23] From [20] thesqueeze damping can be represented by the lump resistanceas

119877119904119902119890119890119911119890 = 31205871205831198861198941199031198774120581 (120573)2ℎ3119899ℎ119900119897119890 (24)

where

120581 (120573) = 41205732 minus 1205734 minus 4 ln (120573) minus 3 (25)

120573 = 119886(1051199031radic2) (26)

where 1199031 represents the distance between center of hole andcenter of membrane and nhole is the total number of holes

For the plate although there are other damping mecha-nisms the squeeze film damping is the most noticeable onceit is presented in MEMS devices [23] In addition CMUTis an acoustic device the loss of acoustic radiation cannotbe ignored [10] According to [21 22] the resistance of theacoustic radiation of the clamped plate is

119877119903119886119889 119901119897119886119905119890 = 120588119888119860 [[1 minus 201198651 (2 (120596119888) radic119860120587)((120596119888) radic119860120587)9 ]]

1198651 (119909) = (1199094 minus 911199092 + 504) 1198691 (119909)+ 14119909 (1199092 minus 18) 1198690 (119909) minus 119909516 minus 1199097768

(27)


where 119869119899(x) is the Bessel function of the first kind of order 119899Thus the damping resistance of the top plate 119877119901 is

119877119901 = 119877119904119902119906119890119890119911119890 + 119877119903119886119889 119901119897119886119905119890 (28)

where Rsqueeze is calculated as in (24) to (26) Rrad plate iscalculated as in (27)

For the equivalent air piston not only the acousticradiation damping but also the viscous damping needs tobe considered When the air pistons vibrate due to theviscosity of the air they are damped by the viscous force atthe side walls of the holes From [24] the resistance of viscousboundary layer per unit area can be expressed as

119877V119894119904119888 = radic 1205881205831198861198941199031205962 (29)

When calculating the viscous damping of the equivalent airpiston the area of viscous boundary layer is taken as the areaof the side walls of the holes and a viscous end correction [19]The resistance of acoustic radiation Rrad air piston is calculatedas the baffled piston source [14]

119877119903119886119889 119886119894119903 119901119894119904119905119900119899 = 119899ℎ119900119897119890120588119888 (1205871198862) (1 minus 21198691 (2120596119886119888)2120596119886119888 ) (30)

Thus the damping resistance of the equivalent air piston is

119877119886 = 119899ℎ119900119897119890 (2120587119886119905 + 41205871198862) 119877V119894119904119888 + 119877119903119886119889 119886119894119903 119901119894119904119905119900119899 (31)

After getting 119880119886 and 119880119901 the complex amplitude of totalvolume velocity 119880 can be written as

119880 = 119860119880119901 + 119878119880119886 (32)

From [21 25] if we consider a CMUT cell as a clamped edgeradiator source the complex amplitude of sound pressure ofthe point at distance 119889 in the far field above the center of themembrane can be calculated as

119901 = 1198941205961205881198802120587119889 (33)

then the sound pressure level (SPL) at this point can becalculated

22 Finite Element Method Model In order to investigate theair-coupled unsealed Helmholtz structural CMUTs and val-idate the analytical model we have developed the finite ele-ment method (FEM) model to simulate the device UnsealedCMUTswould induce squeeze film effect usually the squeezefilm effect in MEMS devices is described by nonlinearcompressible Reynolds equation which is simplified from thecontinuity equation and the Navier-Stokes equation when themodified Reynolds number satisfies [13 20]

Re119898119900119889119894119891119894119890119889 = 120596120588ℎ2120583119886119894119903 ≪ 1 (34)

This condition is not effective for CMUTs which have slightlyhigher gaps or higher operating frequencies Thus we need

to start with the continuity equation and the Navier-Stokesequation to model the squeeze film effect in our unsealedHelmholtz structural CMUTs [13]

The commercial software COMSOLMultiphysics version52a is used to build our FEM model The Electromechani-cal (emi) module and Thermoviscous Acoustics FrequencyDomain (ta) module are used in our FEMmodel of a CMUTcell The governing equations of ta include the Navier-Stokesequation the continuity equation and the energy equationformulated using the Fourier heat law [26]Thus the squeezefilm effect is included and the thermal loss is also taken intoaccount

We use four symmetric distributed holes here to avoidthe damage of the center part of membrane which makesmain contribution to membranersquos vibration Table 1 lists thematerial properties in simulation The 3D model of a quarterof a CMUT cell in octant spherical air domain is built asshown in Figures 3(a) and 3(b) A ldquoperfectly matched layer(PML)rdquo is used to eliminate the acoustic reflection and tosimulate an infinite spread medium In emi module for theldquodomain selectionrdquo it includes all domains The membranethe insulator layer and the support are selected as ldquolinearelastic dielectricrdquo The rim of the membrane the supportand the insulator layer are applied with ldquofixed constraintrdquoboundary condition The substrate can be ignored because ofthe fixed insulator layer For simplification in FEM the topand bottom electrodes can be regarded as infinitely thin [27]thus we directly use the membranersquos top surface as the topelectrode and the bottom electrode is located underneath theinsulator layer A constant DC bias voltage and a harmonicAC voltage (amplitude as 1 of DC voltage) are applied on thetop surface of membrane by the ldquoterminalrdquo with ldquoharmonicperturbationrdquo conditions and the bottom surface of insulatorlayer is set as ldquogroundrdquo The ldquosymmetryrdquo is applied onthe vertical sections of the membrane insulator layer andsupport Other boundary conditions keep the default settingin COMSOL In ta module for the ldquodomain selectionrdquo onlydomains filled with air (cavity holes spherical air domainPML) are selected To couple two modules ldquovelocityrdquo andldquoisothermalrdquo boundary conditions are applied in ta on thetop surface of membrane and inner wall of cavity and of holewhich are interfaces between ta and emi [26] The ldquovelocityrdquoin ta is prescribed as in emi This coupling setting is a two-way coupling which means when the membrane vibrates togenerate sound waves in medium a reaction force would beadded to the membrane [28] Thus we do not need to addother coupling condition in emi module The ldquosymmetryrdquoboundary condition is applied on the vertical sections ofcavity and the vertical sections of octant spherical air andPML Other boundary conditions keep the default settingin COMSOL In the mesh setup the mesh size should besufficiently small to resolve the wavelength by at least 5-6elements per wavelength [29] and the maximum elementsize of the acoustic medium is set as one-sixth of wavelengthof operating frequency Also the ldquoboundary layers meshrdquois needed at the inner walls of both cavity and holes thethickness of ldquoboundary layers meshrdquo is equal to the viscousboundary layer thickness under fundamental resonant fre-quency of CMUTs membrane [26] The meshing details of


Table 1 Material properties used in COMSOL (lowast the factor 06 isfrom [30] others are the preset valuesfunctions in COMSOL)

Parameter ValuePreset function inCOMSOL

Air (used in cavity holes spherical air domain PML)Relative permittivity 1Dynamic viscosity 120583air [Pas] eta(T[1K])Heat capacity at constant pressureCp [J(kgK)]

Cp(T[1K])

Density 120588air [kgm3] rho(pA[1Pa]T[1K])

Thermal conductivity K [W(mK)] k(T[1K])lowastBulk viscosity 120583b [Pas] 06lowasteta(T[1K])Typical Wave Speed for PML [ms] 343

Si (used in membrane support beam)Youngrsquos modulus Y [GPa] 170Poisson ratio 120590 028Density 120588Si [kgm

3] 2329Relative permittivity 117

SiO2 (used in insulator layer)Youngrsquos modulus [GPa] 70Poisson ratio 017Density [kgm3] 2200Relative permittivity 42

vertical section of the cell are shown in Figure 3(c) where theldquoboundary layers meshesrdquo are highlighted The ldquoprestressedanalysisrdquo is applied in calculating the average pressure overdifferent frequencies Firstly a stationary study is used tocalculate the membrane deflection under a certain DC biasvoltage Then an AC voltage is biased on the membrane usingharmonic analysis over a frequency range The pressure iscalculated by averaging the pressure between the membraneand the air medium

In order to make a comparison between the unsealedHelmholtz structural CMUTs and the conventional vacuumsealed CMUTs we also build an FEM model for the latterThemodel building process of the conventional cell is slightlydifferent from the unsealed Helmholtz structural CMUT cellThe solid-air interfaces for the conventional cell only containthe top surface of the membrane thus the ldquovelocityrdquo andldquoisothermalrdquo are only added on the membranersquos top surfaceThe cavity is vacuum thus for a conventional cell we donot take the acoustic field of the cavity into ta modulersquoscalculation Also the ldquoboundary layers meshrdquo is not neededfor the vacuum cavity

3 Results

31 FEMResults We take three (Standard Double andHalf)cases in FEM simulation Table 2 shows the parametersof physical structural dimension of the CMUT cell usedThe main target of this paper is to demonstrate synergyproperties of both the squeeze film effect and the Helmholtzresonant effect in CMUTs In order to fulfill this we set

(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity

Insulator layer Support

Boundary layers mesh

Figure 3 A 3D model in COMSOL (a) a quarter of a CMUT celland spherical air domain (b) details of the unsealed Helmholtzstructural CMUT cell (c) meshing details of the vertical section of(b)

the corresponding parameters and they turn out to be a bitlarge for Standard case In the Standard case the radius 119877and thickness 119905 of membrane are referenced from [12] as arepresentative one The cavity height ℎ is designed to achievea proper h2120590visc ratio to generate the obvious Helmholtzeffect where 120590visc is the thickness of the viscous boundarylayer as we will explain in the discussion section For air-coupled CMUTs designs and fabrications with such largecell dimensions can be found in [31 32] The fabricationmethod of the vented holes can be found in [12 13] In theDouble case the physical dimension parameters are 2 timesof the Standard case In the frequency response the vibrationcoupling characteristic of two resonant peaks ismore obviouswith such large dimension cells because the proportion ofthermal viscous loss energy to the total energy gets lowerwhen the cell dimension increases from Half case to Doublecase as we shall show in discussion section The results of theDouble case with obvious characteristic are helpful to clearly


Table 2 Parameters of CMUT cells in our design

Parameters Double Standard HalfRmembrane radius [120583m] 1000 500 250h cavity height [120583m] 60 30 15t membrane thickness [120583m] 20 10 5aini initial hole radius [120583m] 110 55 275a hole radius [120583m] 40 80 120 160 200 20 40 60 80 100 10 20 30 40 50tins insulator layer thickness [120583m] 6 3 15r1 hole position [120583m] 750 375 1875nhole holes number 4 4 4Vdc DC voltage [V] 6720 3360 1680

investigate the working mechanism of Helmholtz structuralCMUT In the Half case the physical dimension parametersare taken as half of the Standard case As we shall show indiscussion section this case is to investigate influence factorof the Helmholtz effect

For hole radius we use (6) to calculate 119891119901 let 119891ℎ = 119891119901then use (9) (10) and (15) to solve an initial hole radius ainiof each hole as

119886119894119899119894= 096radic119899ℎ119900119897119890120587119902119886 + radic0962119899ℎ1199001198971198901205871199021198862 minus 4 (1 + 119902119886119902119887) (minus119902119886119905)2 (1 + 119902119886119902119887)

(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)

Then choose hole radius around aini In each case in Table 2compared with the no-hole cell the collapse voltage of theCMUTs with 4 holes decreases a little For simplicity the DCvoltage is taken as about 70 of the collapse voltage of theno-hole cell

Firstly the Double case is considered to show thevibration coupling working mechanism of the air-coupledunsealed Helmholtz structural CMUTs

The membrane and air pistonrsquos motions are mainly onvertical direction thus in COMSOL we take the amplitudeof the membranersquos velocity 119880119901 and the amplitude of theair pistonrsquos velocity 119880119886 as the z directional average velocitycomplex amplitude of the membranersquos top surface and of theholersquos top surface respectively

Figure 4 shows magnitudes of 119880119901 and 119880119886 As shown inFigure 4 for unsealed Helmholtz structural CMUTs withdifferent hole sizes as mentioned in methodologies sectionthe magnitudes of 119880119901 and 119880119886 have two resonant frequenciescorresponding to 119891minus = 120596minus2120587 119891+ = 120596+2120587 respectivelyIt is noticeable that between these two resonant peaks thereis a trough whose corresponding frequency is called theantiresonant frequency Referring to [16] this antiresonantfrequency is equal to the resonant frequency of uncoupledHelmholtz resonator 119891ℎ

The FEM complex amplitude of total volume velocity 119880is calculated as (32) Figure 5 shows its magnitude |119880| Also|119880| of the unsealed Helmholtz structural CMUTs has tworesonant frequencies locating at two sides of the peak of thevacuum sealed cell at fundamental resonant frequency 119891119901 =83 kHz

Figure 6 illustrates the air pistonsrsquo and the membranersquoscontributions to the magnitudes of total volume velocity |119880|of the unsealed Helmholtz structural CMUT cells Contri-butions of air pistons and the membrane are represented bythe magnitudes |119878119880119886| and |119860119880119901| respectively In order tomark the fundamental resonant frequency 119891119901 we also addthe magnitudes of total volume velocity of the vacuum sealedcell From Figures 6(a) and 6(b) for the unsealed Helmholtzstructural CMUT cells when 119891ℎ lt 119891119901 the air pistons makemajor contribution to the left peak of |119880| at 119891minus and to theright peak at f + while 119891ℎ gt 119891119901 In simpler terms for theunsealed Helmholtz structural CMUT cells the air pistonsmainly contribute to the resonant peak whose correspondingfrequency is farther from 119891119901 than the other resonant peak

The CMUTs from the Double case to the Half casehave different working frequencies thus at the point withsame distance from the CMUT cells the acoustic attenuationdiffers among cases In order to fairly compare the outputpressure from the Double case to the Half case the SPL istaken at the top surfaces of membrane and hole We firsttake the area average sound pressure of the top surfacesof membrane and hole then calculate the SPL Figure 7shows the SPLs over the membrane and holersquos top surfaceFrom (33) the sound pressure is proportional to U on thisaccount each curve in Figure 7 has two resonant peaks atresonant frequencies of its corresponding curve in Figure 5As shown in Figure 7 compared with the sealed one the SPLsof unsealed Helmholtz structural CMUTs realize the dual-band and extend range of frequency response towards widerfrequencies

Secondly for the Standard case in Table 2 the FEM resultsof SPLs are shown in Figure 8 Two resonant frequenciesdue to the vibration coupling also can be observed for theunsealed cells As we can see from Figure 8 for SPLs thebandwidths of unsealed Helmholtz structural CMUT cellsare obviously larger than the conventional sealed CMUT cellHowever compared with the sealed cell the energy loss inthe form of squeeze damping loss viscous loss and thermal


01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)

FEM a=40um|Ua||Ua|FEM a=80um

FEM a=120um|Ua|FEM a=160um|Ua|FEM a=200um|Ua|

30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)

FEM a=40umFEM a=80umFEM a=120um|Up|FEM a=160um|Up|FEM a=200um|Up|no hole|Up|

30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)

Figure 4 FEM results of the Double case (a) magnitude of complex amplitude of air piston velocity (b) magnitude of complex amplitudeof membrane velocity

30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|

FEM a=80umFEM a=120um|U|FEM a=160um|U|FEM a=200um|U|FEM no hole

Figure 5 FEM results of the Double case magnitude of complexamplitude of total volume velocity

loss is introduced in the cavity and holes of the unsealedHelmholtz structural CMUT cell Under the same 119881119889119888 and119881119886119888 for the sealed cell the electrical energy is convertedinto the acoustic energy while for the unsealed cell theelectrical energy is converted into the acoustic energy andthe energy loss in the cavity and holes Thus the pressurevalues of the Helmholtz structural CMUT cells are lowerthan the sealed one Referring to [33] we define the SPL-bandwidth product as bandwidth [MHz]timespeak value of SPL[dB] both bandwidth and pressure values performance ofthe CMUTs can be reflected by this product We calculatethe 3-dB fractional bandwidth and SPL-bandwidth product

of the CMUT cells in Figure 8 the results are listed inTable 3

As shown in Table 3 the 119891119901 is calculated from (6) andthe 119891ℎ is calculated from (9) and (10) when 119891ℎ is closeto 119891119901 the unsealed Helmholtz structural CMUT cell witha = 60 120583m achieves a significant improvement around 35times of the conventional sealed cell in both 3-dB fractionalbandwidth and SPL-bandwidth product Other unsealeddimensions also have performance enhancement comparedwith the conventional sealed one Therefore compared withthe conventional sealed cell unsealed Helmholtz structuralCMUT cells not only enhance the bandwidth of SPL but alsokeep the pressure value at a reasonable level

Last for the Half case in Table 2 the FEM resultsof the SPLs over the top surfaces of the membrane andhole are shown in Figure 9 and the corresponding 3-dBfractional bandwidth and the SPL-bandwidth product arelisted in Table 4 The characteristic of two resonant peaksis not so obvious as the Double case and the Standard caseHowever the significant enhancement around 17 times of theconventional sealed cell still is achieved while119891ℎ is close to119891119901(a = 30120583m cell) Unsealed Helmholtz structural CMUT cellsstill have a performance enhancement over the conventionalsealed CMUT cell

As shown above in Tables 3 and 4 compared with theconventional sealed CMUT cell the air-coupled unsealed 4-holeHelmholtz structural CMUT cells achieved the enhance-ment of both 3-dB fractional bandwidth and SPL-bandwidthproduct and the significant enhancement was achieved while119891ℎ is close to 11989111990132 Analytical Results Compared with FEM Results In orderto validate our analytical model we make a comparisonbetween analytical results and FEM results from all cases Forthe analytical results we first use (15) to (22) (28) and (31)


0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ

FEM a=40umFEM a=40umFEM a=40um

|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)

Figure 6 FEM results of the Double case (a) |119878119880119886| |119860119880119901| |119880| of a=40120583m dimension (119891h lt 119891p) (b) |119878119880119886| |119860119880119901| |119880| of a=160120583m dimension(119891h gt 119891p)

Table 3 3-dB fractional bandwidth and SPL-bandwidth product of the Standard case as shown in Figure 8

Hole radius [120583m] a=20 a=40 a=60 a=80 a=100 no holef h [kHz] 923 1385 1765 2113 2447 119891119901=1675Separation between f p and f h [kHz] 752 29 9 438 7723-dB fractional bandwidth 48 116 357 73 38 09SPL-bandwidth product 122 288 801 151 077 023

30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)

FEM a=40umFEM a=80umFEM a=120umFEM a=160umFEM a=200umFEM nohole

Figure 7 FEM results of theDouble case soundpressure level (SPL)over the membrane and holersquos top surface (dB)

to calculate values of basic parameters then we use (6) toget 120596119901 and we use (9) and (10) to get 120596ℎ After that we use(11a) to (13) to get 119880119901 and 119880119886 then we substitute back (32) toget 119880 and use (33) to get 119901 at a point above (d = 3600120583min the Double case d =1800 120583m in the Standard case d =900120583m in the Half case) the center of the membrane Then

95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)

FEM a=20umFEM a=40umFEM a=60umFEM a=80umFEM a=100umFEM no hole

80 100 120 140 160 180 200 220 240 260 280 300 320

Figure 8 FEM results of the Standard case sound pressure level(SPL) over the membrane and holes (dB)

we calculate SPL at this point In this section the FEM SPLresults also are calculated at the same corresponding pointabove the center of the membrane The comparison resultsare shown in Figure 10

One point we would like to mention in (20) and (27)we calculate the acoustic radiation impedance of the CMUT


Table 4 3-dB fractional bandwidth and SPL-bandwidth product of the Half case as shown in Figure 9


150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)


Figure 9 FEM results of the Half case sound pressure level (SPL)over the membrane and holes (dB)

membrane by the formulas of the baffled clamped platesource From [22] the piston plate radiation and the clampedplate radiation are almost same for kRsource lt1 where 119896 is thewave number and Rsource is the source radius In our designthe value of kRsource is small thus we can also use baffledpiston source to calculate the acoustic radiation impedanceof the CMUT membrane the analytical results are almostsame by using these two ways to calculate acoustic radiationimpedance of the CMUT membrane

In order to evaluate the similarity between the analyticalSPL results and the FEM SPL results we calculate the nor-malized mean square error (NMSE) between our analyticalresults and FEM results [34]

119873119872119878119864 V119886119897119906119890 = 1 minus 10038171003817100381710038171003817997888rarr119909 119903119890119891 minus 997888rarr11990910038171003817100381710038171003817210038171003817100381710038171003817997888rarr119909 119903119890119891 minus 119898119890119886119899 (997888rarr119909 119903119890119891)100381710038171003817100381710038172 (38)

The NMSE values vary between negative infinity (not fit)and 1 (perfect fit) In each subgraph in Figure 10 we takeFEM results and analytical results as997888rarr119909 119903119890119891 and997888rarr119909 respectivelyThe calculated NMSE values in Table 5 indicate our analyticalresults have a good matching with FEM results

The deviations between analytical results and FEM resultscould be caused by the simplifications of the analyticalmodel such as negligence of themembranersquos deformation andassumption as a plate with uniform displacement In (22) theanalytical driving force is only taken as the harmonic part

Table 5 Normalized mean square error (NMSE) values betweenanalytical results and FEM results in Figure 10

Double case a=80120583m a=120120583mNMSE 09173 09347Standard case a=40120583m a=60120583mNMSE 09355 09513Half case a =20120583m a=30120583mNMSE 09523 09689

of the electrical force In (28) and (31) the loss taken intoconsideration is only the squeezing viscous and radiationloss not including the thermal loss which is included inthe FEM model And in the analytical model we use theequivalent air piston instead of four small air pistons inthe holes thus the interaction among holes is neglectedCompared with the no-hole cell the membrane stiffnessand the uncoupled natural resonant frequency of the 4-hole CMUT cell decrease In [15] the natural resonantfrequency of the membrane with holes was approximatedby the formula of the no-hole membrane For our designusing (6) of the no-hole membrane to approximate thenatural resonant frequency of the 4-hole membrane wouldnot have a significant influence of the analytical resultsIn the future work the accurate formula of the naturalresonant frequency of the membrane with holes will beinvestigated

However this analytical model gives the explanation ofthe most significant vibration coupling mechanism of the air-coupled unsealed Helmholtz structural CMUTs and can beused to predict the characteristics such as |119880| SPL and f +and 119891minus of this kind of CMUTs during the design processFigure 11 shows the flow chart on how the analytical modelcan be used to aid the design ofHelmholtz structural CMUTsIn the design phase of Helmholtz structural CMUTs for themembrane with inputted thickness 119905 and radius R we firstlyselect an ℎ value within the interval [4120590visc ℎmax] where thelower limit 4120590visc is for the obvious Helmholtz resonanceand the consideration of the limitation of DC voltage andmaximum allowed gap height is included in ℎmax Thensweep hole radius 119886 around the calculated initial hole radiusaini to get the point SPL and perform the FEM simulationof the dimension neighbored around the hole radius withsignificant performance enhancement to search for moreaccurate fine results If there is no significant performanceenhancement for all of the swept hole radii then we need toreselect the ℎ value and sweep hole radius 119886 around the newcalculated aini to find performance enhanced cell dimension


0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

FEM a=80umanalytical a=80um






0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)

Figure 10 Comparison of analytical and FEM results of SPL at a point above the center of the membrane

If there is no proper h we need to go back to reinput 119905 and119877 Using this design process the analytical model can helpus to initially pick out the cell dimensions with performanceenhancement Then we only need to perform the FEMsimulations of these dimensions to check more accurateresults This simplifies the design process and shortens thedesign period saving a lot of computation time with ourreasonable guesstimates of analytical design procedure

4 Discussion

41 Explanation of the Decreased Phenomenon of the TwoResonant Peaks In this section we make a comparison ofthe FEM results of SPLs from the Double case to the Halfcase and give an explanation of the decreased phenomenonof the two resonant peaks As we can see from Figure 8 forthe SPL of the unsealed Helmholtz structural CMUT cells in


Input t R Input material property parameters of

Silicon (membrane material) and Air

Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)

Choose a with a defined step size around

(start from a=05gradually increase)

Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes

FEM simulation of the CMUT cell of this dimension to see more

accurate results

Set Solve

Calculate basic parameters Yes]

No

Yes

NoSPL level ( difference

within 1dB )The SPL of the

Two resonant peaks are at same

trough between two resonant peaks is lower within

6dB than the peak SPL

hwithin the interval

a within the interval

[05 15

Calculate p

Calculate isc

Choose ℎge4iscwith a

(start from ℎ=4isc)

[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p

(20 times 10minus6))

S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D

Figure 11 The flow chart of using analytical model to aid the design process of Helmholtz structural CMUTs

the Standard case the left peaks of cells with 119891ℎ lt 119891119901 andthe right peaks of cells with 119891ℎ gt 119891119901 are not as evident asthe Double case in Figure 7 For the Half case in Figure 9for the unsealed Helmholtz structural CMUT cells the peaksmajorly contributed by air pistons appear further weakly andeven almost disappearThis is due to the thermal viscous losswhich is caused by viscosity and heat conduction

When sound propagates in structures with small geo-metric dimensions the interaction between air and solidboundaries gives rise to the viscous boundary layer and thethermal boundary layer In these two boundary layers theloss is much greater than in the free field which is far awayfrom the solid boundaries The thicknesses of these twoboundary layers are calculated as [24]

120590V119894119904119888 = radic 2120583119886119894119903120588119886119894119903120596 asymp 22 times 10minus3radic119891 (39)

120590119905ℎ119890119903119898 = radic 2119870120588119886119894119903119862119901120596 asymp 25 times 10minus3radic119891 (40)

For viscous boundary layer as Figure 12 shows we canregard the flow in the cavity and the holes as shear flow itcan be divided into two parts the part within 120590visc is heavilydamped and with great mechanical loss and the other partfar from 120590visc is free and lossless As listed in Table 6 theratio ℎ2120590visc reduces from the Double case to the Half caseTherefore for the whole flow in the cavity and the holesthe proportion of the part with heavy loss increases and theproportion of the lossless part decreases That means moreportion of the energy is dissipated by the boundary layers andportion of the energy provided for the air pistonsrsquo vibrationgetsweaker as a consequence the peakmostly contributed byair pistons gets weaker For thermal boundary layer also theproportion of the total energy to the thermal loss energy canbe reflected by the ratio h2120590therm For our designed CMUTsparameters the thermal loss can be neglected comparedwith the squeeze film loss and the radiation loss Thus inour analytical model we did not take the thermal loss intoaccount

We have also simulated other two cases whose dimen-sions are 10 times and 01 times of the Standard case in Table 2


Table 6 Ratio of cavity height to 2120590visc taking f as f p in (6)

Double case Standard case Half caseh2120590visc 394 279 197

Viscous boundary layerFree flowDamped flow

Figure 12 Illustration of viscous boundary layer and flow in cavityand holes

The results are in accordance with our explanation aboveFor the 10 times case the two resonant peaks are obviousBut for the 01 times case the Helmholtz resonant effecttends to disappear only one resonant peak is observed in theSPL frequency response We also did FEM simulations andanalytical analyses for a R = 650 120583m t = 10 120583m and h = 30120583m4-hole cell with different hole radii Compared with the cellin Standard case the Helmholtz effect gets weaker because ofthe decreased h2120590visc ratio Thus when we sweep hole radiifrom 25120583m to 45 120583m around aini = 36120583m the two resonantpeaks cannot bewithin 3 dB level and cannot achieve the 3 dBbandwidth improvement like a = 60 120583m cell in Figure 8

42 Contribution of Helmholtz Resonant Effect to the Out-put Enhancement of the 4-Hole Cell Here we check thebandwidth and SPL-bandwidth product enhancement of theunsealed Helmholtz structural CMUTs is indeed affected bythe Helmholtz resonator although the vibration couplingfeature of two resonant peaks is not obvious in Figures 8and 9 The hole area ratio is defined as the total hole areadivided by the total membrane area ie nhole120587a2120587R2 Wesimulate an unsealed cell with 68 holes uniformly distributedon the membrane and having the same hole area ratio asthe cell (a=60120583m 4 holes) with significant enhancement inthe Standard case Other dimension parameters of this 68-hole cell remain the same as the Standard case in Table 2Thus from (9) (10) and (15) for the 4-hole cell and the68-hole cell they have the same value of the uncoupledHelmholtz resonant frequency 119891ℎ The simulation resultsare shown in Figure 13 We can see from Figure 13 as weexplained in Figure 12 in spite of the same 119891ℎ there ismore air-solid contact area contributed by holesrsquo side wallin the 68-hole cell Hence there are more lossy flows whichweaken the vibration of air pistons of Helmholtz resonatorTherefore the performance of 68-hole CMUTwith decreasedHelmholtz resonator effect is much less than the 4-holeonersquos

We also simulate another 68-hole cell which has the samehole area ratio as the cell (a=30120583m 4 holes) with a significantenhancement in the Half case other dimension parameters

0 100 200 300 4005060708090

100110120130140150160

68 holes FBW37 SPL-BWP081


no hole FBW09 SPL-BWP023Standard case

frequency (kHz)

FEM a=60um 4holes FEM a=145um 68holes FEM no hole

the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r

Figure 13 FEM results of the Standard case SPL comparison oftwo cells with same hole area ratio a=60120583m 4 holes amp a=145120583m 68holes (FBW 3-dB fractional bandwidth SPL-BWP SPL-bandwidthproduct)

0 200 400 600 8005060708090

100110120130140150160



no hole FBW13 SPL-BWP067

frequency (kHz)

FEM a=30um 4holesFEM a=725um 68holesFEM no hole

Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case

Figure 14 FEM results of the Half case SPL comparison of twocells with same hole area ratio a=30120583m 4 holes amp a=725120583m 68holes (FBW 3-dB fractional bandwidth SPL-BWP SPL-bandwidthproduct)

of this 68-hole cell are the same as the Half case in Table 2 Asshown in Figure 14 the 4-hole cell still performs better thanthe 68-hole one which means Helmholtz resonant effect stillhas a great influence for the performance improvement of the4-hole unsealed Helmholtz structural CMUT cells in the Halfcase although the characteristic of two resonant peaks is notobvious

43 Range of Effectiveness for Analytical Model From dis-cussion above when the h2120590visc ratio decreases or the holenumber increases under same hole area ratio the Helmholtzresonant effect gets weaker and even disappears For ouranalytical model the existence of Helmholtz resonant effectis a basic assumption In Figure 11 when choosing thecavity height h the condition hge4120590visc is an FEM empirical


120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)

FEM a=40umFEM a=45umFEM a=50umFEM a=55umFEM a=60umFEM a=65umFEM a=70umFEM a=75umFEM a=80um

Figure 15 FEM results of the Standard case sound pressure level(SPL) over the membrane and holes (dB) hole radius changes from40120583m to 80 120583m with 5 120583m step size

value that the Helmholtz resonant effect is strong enoughto benefit the bandwidth of the CMUT device Our ana-lytical model is effective under these situations where theHelmholtz resonant effect is strong For smaller h2120590visc ratiothe Helmholtz resonant effect tends to disappear and ouranalyticalmodel loses efficacy Also for 4-hole cells in Figures13 and 14 the Helmholtz resonant effect is strong and ouranalytical model is effective For 68-holes cells the analyt-ical model loses efficacy In the future work the researchattempt is to investigate the effective range of our analyticalmodel

44 The ldquoBestrdquo Hole Radius for Bandwidth EnhancementFrom Tables 3 and 4 we can see when 119891ℎ is close to 119891119901 theunsealed Helmholtz structural CMUT achieves a significantoutput pressure bandwidth performance enhancement Wecan see a tendency that when 119891ℎ = 119891119901 the improvementshould achieve the ldquobestrdquo We use 5120583m step size to simulateStandard case in Figure 8 with hole radius changes from40120583m to 80 120583m the results of SPL over the membrane andholes are shown below As Figure 15 shows the a=55120583ma=60120583m and a=65120583m cells have larger 3-dB bandwidththan other dimensions The 3-dB fractional bandwidth are294 357 and 297 for a=55 120583m 60 120583m and 65 120583mrespectively The calculated aini which let 119891ℎ = 119891119901 is 55120583mhowever its fractional bandwidth enhancement is not theldquobestrdquo one For the ldquobestrdquo fractional bandwidth enhancementwith a=60 um as we described above the 119891ℎ is close to119891119901 Up to now we cannot give a quantitative analysis forhow close are the 119891ℎ and 119891119901 for the ldquobestrdquo bandwidthperformance enhancement This still needs to be furtherinvestigated

5 Conclusions

In this paper we present the air-coupled unsealed Helmholtzstructural CMUTs which utilize both Helmholtz resonanteffect and squeeze film effect to enhance the devicersquos outputpressure bandwidth performance in transmit modeWe havedeveloped an analytical model to explain the vibration cou-pling between the vibration ofmembrane and the vibration ofair pistons in the holes which is the basicworkingmechanismof the Helmholtz structural CMUTs In order to validate theanalytical model we use FEM model to simulate three casesof the 4-hole unsealed Helmholtz structural CMUT cellsFrom the FEM results for the Double case the two resonantpeaks in the frequency response clearly show the vibrationcoupling working mechanism of the Helmholtz structuralCMUT cell For an unsealed Helmholtz structural CMUTcell with various hole radii compared with the conventionalsealed one the improvements of 3-dB fractional SPL band-width and SPL-bandwidth product are around 35 times inthe Standard case and 17 times in the Half case which areachieved with the hole radius which let 119891ℎ being close to 119891119901Compared with FEM results within reasonable deviationsthe analytical model can predict the characteristics such as|119880| SPL and f + and 119891minus of the unsealed Helmholtz structuralCMUTs In the design phase of Helmholtz structural CMUTsthis analytical model can help to initially select the celldimensions with output pressure bandwidth performanceenhancement thus simplifying the design process and short-ening the design period The smaller h2120590visc ratio or themore holes under the same hole area ratio the weaker thepeak majorly contributed by air pistons which means theweaker Helmholtz resonant effect Our analytical model losesefficacy when the Helmholtz effect tends to disappear andthe exact effective range of our analytical model still needs tobe further investigated In summary the air-coupled 4-holeHelmholtz structural CMUT cell which is designed based onthe vibration coupling mechanism can greatly enhance theoutput pressure bandwidth and keep the output pressure at areasonable levelThe analytical model can simplify the designprocess of this kind of CMUTs

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Small part of this initial work was presented (but notpublished) as a poster at MDBS-BHE 2017 The 11th IEEE-EMBS International Summer School and Symposium onMedical Devices and Biosensors This work was supportedby the Science and Technology Development Fund of Macau(FDCT) under 0932015A3 0882016A2 the University of


Macau under Grants MYRG2018-00146-AMSV MYRG2016-00157-AMSV MYRG2015-00178-AMSV the Nature ScienceResearch Project of Lingnan Normal University under GrantZL1901 and Science and Technology Research Project ofZhanjiang City under Grant 2018B01003

References

[1] I Ladabaum X Jin H T Soh A Atalar and B T Khuri-YakubldquoSurface micromachined capacitive ultrasonic transducersrdquoIEEE Transactions on Ultrasonics Ferroelectrics and FrequencyControl vol 45 no 3 pp 678ndash690 1998

[2] A Caronti G Caliano R Carotenuto et al ldquoCapacitive micro-machined ultrasonic transducer (CMUT) arrays for medicalimagingrdquo Microelectronics Journal vol 37 no 8 pp 770ndash7772006

[3] O Oralkan A S Ergun C-H Cheng et al ldquoVolumetric ultra-sound imaging using 2-D CMUT arraysrdquo IEEE Transactions onUltrasonics FerroelectricsampFrequencyControl vol 50 no 11 pp1581ndash1594 2003

[4] S Takahashi and H Ohigashi ldquoUltrasonic imaging using air-coupled P(VDFTrFE) transducers at 2 MHzrdquo Ultrasonics vol49 no 4-5 pp 495ndash498 2009

[5] K-N Huang and Y-P Huang ldquoMultiple-frequency ultrasonicdistance measurement using direct digital frequency synthesiz-ersrdquo Sensors and Actuators A Physical vol 149 no 1 pp 42ndash502009

[6] K Lindstrom L Mauritzson G Benoni P Svedman and SWillner ldquoApplication of air-borne ultrasound to biomedicalmeasurementsrdquo Medical and Biological Engineering and Com-puting vol 20 no 3 pp 393ndash400 1982

[7] M Kupnik A Schroder and M Groschl ldquoPS-16 adaptiveasymmetric double-path ultrasonic transit-time gas flowmeterrdquoin Proceedings of the IEEE International Ultrasonics Symposium(IUS rsquo06) pp 2429ndash2432 Vancouver BC Canada October2006

[8] W Manthey N Kroemer and V Magori ldquoUltrasonic transduc-ers and transducer arrays for applications in airrdquo MeasurementScience and Technology vol 3 no 3 pp 249ndash261 1992

[9] M Kupnik M-C Ho S Vaithilingam and B T Khuri-YakubldquoCMUTs for air coupled ultrasoundwith improved bandwidthrdquoin Proceedings of the IEEE International Ultrasonics Symposium(IUS rsquo11) pp 592ndash595 Orlando Fla USA October 2011

[10] S Hansen A Turo F Degertekin and B Khuri-Yakub ldquoChar-acterization of capacitive micromachined ultrasonic transduc-ers in air using optical measurementsrdquo in Proceedings of theIEEE International Ultrasonics Symposium (IUS rsquo00) pp 947ndash950 San Juan Puerto Rico USA October 2000

[11] N Apte K K Park and B T Khuri-Yakub ldquoExperimental eval-uation of CMUTs with vented cavities under varying pressurerdquoin Proceedings of the IEEE International Ultrasonics Symposium(IUS rsquo13) pp 1724ndash1727 Prague Czech Republic July 2013

[12] N Apte K K Park A Nikoozadeh and B T Khuri-YakubldquoBandwidth and sensitivity optimization in CMUTs for air-borne applicationsrdquo in Proceedings of the IEEE InternationalUltrasonics Symposium (IUS rsquo14) pp 166ndash169 Chicago Ill USASeptember 2014

[13] N Apte K K Park A Nikoozadeh and B T Khuri-YakubldquoEffect of fluid losses and acoustic resonances in CMUTswith vented cavitiesrdquo in Proceedings of the 18th InternationalConference on Solid-State Sensors Actuators and Microsystems

(Transducers 2015) pp 682ndash685 Anchorage Alaska USA June2015

[14] L E Kinsler A R Frey A B Coppens and J V SandersFundamentals of Acoustics John Wiley amp Sons Inc 1999

[15] A Octavio Manzanares and F Montero De Espinosa ldquoAir-coupled MUMPs capacitive micromachined ultrasonic trans-ducers with resonant cavitiesrdquo Ultrasonics vol 52 no 4 pp482ndash489 2012

[16] O Christensen and B B Vistisenm ldquoSimple model for low-frequency guitar functionrdquoThe Journal of the Acoustical Societyof America vol 68 no 3 pp 758ndash766 1980

[17] V Dickason Loudspeaker Design Cookbook Audio AmateurPubns 7th edition 2005

[18] W Zhang H Zhang S Jin and Z Zeng ldquoA two-dimensionalCMUT linear array for underwater applications directivityanalysis and design optimizationrdquo Journal of Sensors vol 2016Article ID 5298197 8 pages 2016

[19] U Ingard ldquoOn the theory and design of acoustic resonatorsrdquoThe Journal of the Acoustical Society of America vol 25 no 6pp 1037ndash1061 1953

[20] N Apte Capacitive micromachined ultrasonic transducers withvented cavities [PhD thesis] Stanford University 2015

[21] A Unlugedik A S Tasdelen A Atalar and H KoymenldquoDesigning transmitting CMUTcells for airborne applicationsrdquoIEEE Transactions on Ultrasonics Ferroelectrics amp FrequencyControl vol 61 no 11 pp 1899ndash1910 2014

[22] H K Oguz S Olcum M N Senlik V Tas A Atalar and HKoymen ldquoNonlinear modeling of an immersed transmittingcapacitive micromachined ultrasonic transducer for harmonicbalance analysisrdquo IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control vol 57 no 2 pp 438ndash447 2010

[23] R Pratap S Mohite and A K Pandey ldquoSqueeze film effects inMEMSdevicesrdquo Journal of the Indian Institute of Science vol 87no 1 pp 75ndash94 2007

[24] U Ingard Notes on Acoustics (Physics) Infinity Science Press2008

[25] H K Oguz A Atalar and H Koymen ldquoEquivalent circuit-based analysis of CMUT cell dynamics in arraysrdquo IEEE Trans-actions on Ultrasonics Ferroelectrics and Frequency Control vol60 no 5 pp 1016ndash1024 2013

[26] COMSOL 52a ldquoAcoustic module users guide thermoviscousacoustics interfacesrdquo

[27] Y Yu S H Pun P U Mak et al ldquoDesign of a collapse-modeCMUT with an embossed membrane for improving outputpressurerdquo IEEE Transactions on Ultrasonics Ferroelectrics ampFrequency Control vol 63 no 6 pp 854ndash863 2016

[28] ldquoAxisymmetric condensermicrophnerdquo inCOMSOLapplicationgallery httpswwwcomsolnlmodelaxisymmetric-condens-er-microphone-12377

[29] COMSOL 52a ldquoAcoustics model library manual piezoacoustictransducerrdquo

[30] D T Blackstock Fundamentals of Physical Acoustics JohnWileyand Sons 2000

[31] K K Park and B T Khuri-Yakub ldquo3-D airborne ultrasoundsynthetic aperture imaging based on capacitive micromachinedultrasonic transducersrdquoUltrasonics vol 53 no 7 pp 1355ndash13622013

[32] M Ho M Kupnik K K Park and B T Khuri-Yakub ldquoLong-term measurement results of pre-charged CMUTs with zeroexternal bias operationrdquo inProceedings of the IEEE International


Ultrasonics Symposium (IUS rsquo12) pp 89ndash92 Dresden GermanyOctober 2012

[33] S Olcum M N Senlik and A Atalar ldquoOptimization of thegain-bandwidth product of capacitive micromachined ultra-sonic transducersrdquo IEEE Transactions on Ultrasonics Ferro-electrics and Frequency Control vol 52 no 12 pp 2211ndash22192005

[34] httpswwwmathworkscnhelpidentrefgoodnessoffithtml

International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

RoboticsJournal of

Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom


(a)

Vdc

Vac

membrane

substrate

R

t

h

top electrode2a

cavityhole holeinsulator layer

bottom electrode

support supportair pistons

d

Ｌ1

ＮＣＨＭ

(b)

Figure 1 (a) A 4-hole unsealed circular CMUT cell with Helmholtz resonator structure (b) Cross-sectional diagram of this cell along dashedline of (a)

needed for applications such as echo profile evaluation anddistance measurement with high resolution [8] However theconventional vacuum sealed air-coupled CMUTs still own anarrow bandwidth For instance for the CMUT array withonly one cell radius the 6-dB fractional bandwidth is about1 [9] To overcome this unsealed CMUTs were proposedto achieve wider bandwidth by utilizing the squeeze filmeffects [10ndash13] For unsealed CMUTs the vented cavity notonly introduces the squeeze film effects but also constructs aHelmholtz resonator with the vented holes The Helmholtzresonator comprised a cavity with an opening denoted asldquoneckrdquo When an excitation with a specific frequency rangeis applied a volume of air in the neck would vibrate due tothe elasticity of the air inside the cavity and thus radiatessound waves into surroundings This is called Helmholtzresonant effect [14] The Helmholtz resonant effect can alsoexert an influence on the device performance this phe-nomenon was reported but there was a lack of analyticalexplanation in [11] In [15] the Helmholtz resonant effectwas used to enlarge the bandwidth of the CMUTrsquos out-put but in its FEM simulation the squeeze film dampingwas neglected Actually with proper adjustment of volumeratio between vented hole and cavity the performance ofthe device would be benefited by the vibration couplingbetween the air pistons vibration system of Helmholtzresonator and the membrane vibration system of CMUTcell This vibration coupling was widely used to enhancethe low frequency response of musical instrument such asacoustic guitar [16] and vented-box loudspeaker systems[17]

In this paper we present an air-coupled unsealed CMUTcell with Helmholtz resonant cavity which utilizes both thesqueeze film effect and Helmholtz resonant effect to enhancethe output pressure bandwidth of the device For conve-nience we hereafter call it as Helmholtz structural CMUTWe propose an analytical model to explain the vibrationcoupling working mechanism of this kind of CMUT and aidthe design process of this kind of CMUT In order to validatethis analytical model and provide deeper investigation on thiskind of Helmholtz structural CMUT the FEM model is builtwith COMSOL Multiphysics version 52a (COMSOL Inc

Stockholm Sweden) to simulate three cases of 4-hole CMUTswith different hole radii

2 Methodologies

Figure 1(a) shows an air-coupled unsealed Helmholtz struc-tural CMUT cell with 4 holes in its membrane the cross-sectional illustration of this CMUT cell along the red dashedline in (a) is shown in Figure 1(b) Major components ofthis CMUT cell are a circular membrane with radius 119877 andthickness t a cavity with height h holes with radius a thethin metal layer as full top electrode and one insulator layerwith thickness tins to avoid short circuit between electrodesThe bottom electrode is located between the insulator layerand the substrate The rim of the membrane is clampedby support Using Helmholtz resonant effect and squeezefilm effect to enhance bandwidth several circular holes needto be punched through the membrane These holes andthe cavity form a Helmholtz resonator If a DC voltage isapplied between top and bottom electrodes the membranewill be activated by the electrostatic force induced by thevoltage and the restoring spring force due to the membranersquosstiffness When these two forces are equal the membrane willreach an equilibrium position Then when a small amplitudeAC voltage is superimposed the unbalance of these twoforces will be caused by the small perturbation of theelectrostatic force which leads to the membranersquos vibrationThe membranersquos vibration will drive the air pistons in theholes to vibrate due to the elastic property of the air insidethe cavity The vibration of the air pistons will also inducea pressure change in the cavity which leads to a force actedon the membrane Therefore the membranersquos vibration andthe air pistonsrsquo vibration interact with each other During theoperation of this air-coupled unsealed Helmholtz structuralCMUT cell the ultrasonic wave will be generated by bothvibrations of the membrane and the air pistons

21 Lumped-Parameter Analytical Model As Figure 2 showsinspired by the simple model of acoustic guitar [16] theHelmholtz structural CMUT cell can be simplified as aHelmholtz resonatorwith amovable top plateTheHelmholtz


V

SA

+

-

F+

-

+

-

lma

xa

mp

kp

xp











11989811990111988921199091199011198891199052 = 119865 minus 119877119901119889119909119901119889119905 minus (119896119901 + 1205831198602) 119909119901 (4a)

11989811988611988921199091198861198891199052 = minus119877119886 119889119909119886119889119905 minus 1205831198782119909119886 (4b)


120596119901 = radic (119896119901 + 1205831198602)119898119901 (5)


120596119901 = 2120587119891119901 = 2120587 sdot 047 1199051198772radic 119884[120588119898 (1 minus 1205902)] (6)


120596ℎ = 2120587119891ℎ = radic 1205831198782119898119886 (7)


120596ℎ = 119888radic 119878119881119897 (8)


120596ℎ = 119888radic 119878[119881 (119897 + 120575)] (9)





119880119901 = 1198941205961198650 [(120596ℎ2 minus 1205962) + 119894120596120574119886]119898119901119863 (11a)

119880119886 = minus1198941205961198650119860120596ℎ2119898119901119878119863 (11b)



120574119886 = 119877119886119898119886 120574119901 = 119877119901119898119901

120596119901ℎ4 = 120596ℎ212059611988621205961198862 = 1205831198602119898119901

(13)






119860 = 1205871198772 minus 119878 (16)

119881 = 1205871198772ℎ (17)

119898119901 = 18119860119905120588119898 + 119898119903119901 (18)


119898119903119901 = minus120588119888119860120596 [[201198652 (2 (120596119888) radic119860120587)




119898119903119886 = 119899ℎ119900119897119890120588119888 (1205871198862)120596 [21198671 (2 (120596119888) 119886)(2 (120596119888) 119886) ] (21)


1198650 = 10038161003816100381610038161198651198861198881003816100381610038161003816 = 1205760119860119881119889119888119881119886119888ℎ2 (22)



120590119904119902119890119890119911119890 = 121205831198861198941199031205961198772119875119886ℎ2 (23)


119877119904119902119890119890119911119890 = 31205871205831198861198941199031198774120581 (120573)2ℎ3119899ℎ119900119897119890 (24)

where


120573 = 119886(1051199031radic2) (26)





(27)



119877119901 = 119877119904119902119906119890119890119911119890 + 119877119903119886119889 119901119897119886119905119890 (28)



119877V119894119904119888 = radic 1205881205831198861198941199031205962 (29)


119877119903119886119889 119886119894119903 119901119894119904119905119900119899 = 119899ℎ119900119897119890120588119888 (1205871198862) (1 minus 21198691 (2120596119886119888)2120596119886119888 ) (30)


119877119886 = 119899ℎ119900119897119890 (2120587119886119905 + 41205871198862) 119877V119894119904119888 + 119877119903119886119889 119886119894119903 119901119894119904119905119900119899 (31)


119880 = 119860119880119901 + 119878119880119886 (32)


119901 = 1198941205961205881198802120587119889 (33)



Re119898119900119889119894119891119894119890119889 = 120596120588ℎ2120583119886119894119903 ≪ 1 (34)









Cp(T[1K])








3 Results


(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity











(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



V

SA

+

-

F+

-

+

-

lma

xa

mp

kp

xp











11989811990111988921199091199011198891199052 = 119865 minus 119877119901119889119909119901119889119905 minus (119896119901 + 1205831198602) 119909119901 (4a)

11989811988611988921199091198861198891199052 = minus119877119886 119889119909119886119889119905 minus 1205831198782119909119886 (4b)


120596119901 = radic (119896119901 + 1205831198602)119898119901 (5)


120596119901 = 2120587119891119901 = 2120587 sdot 047 1199051198772radic 119884[120588119898 (1 minus 1205902)] (6)


120596ℎ = 2120587119891ℎ = radic 1205831198782119898119886 (7)


120596ℎ = 119888radic 119878119881119897 (8)


120596ℎ = 119888radic 119878[119881 (119897 + 120575)] (9)





119880119901 = 1198941205961198650 [(120596ℎ2 minus 1205962) + 119894120596120574119886]119898119901119863 (11a)

119880119886 = minus1198941205961198650119860120596ℎ2119898119901119878119863 (11b)



120574119886 = 119877119886119898119886 120574119901 = 119877119901119898119901

120596119901ℎ4 = 120596ℎ212059611988621205961198862 = 1205831198602119898119901

(13)






119860 = 1205871198772 minus 119878 (16)

119881 = 1205871198772ℎ (17)

119898119901 = 18119860119905120588119898 + 119898119903119901 (18)


119898119903119901 = minus120588119888119860120596 [[201198652 (2 (120596119888) radic119860120587)




119898119903119886 = 119899ℎ119900119897119890120588119888 (1205871198862)120596 [21198671 (2 (120596119888) 119886)(2 (120596119888) 119886) ] (21)


1198650 = 10038161003816100381610038161198651198861198881003816100381610038161003816 = 1205760119860119881119889119888119881119886119888ℎ2 (22)



120590119904119902119890119890119911119890 = 121205831198861198941199031205961198772119875119886ℎ2 (23)


119877119904119902119890119890119911119890 = 31205871205831198861198941199031198774120581 (120573)2ℎ3119899ℎ119900119897119890 (24)

where


120573 = 119886(1051199031radic2) (26)





(27)



119877119901 = 119877119904119902119906119890119890119911119890 + 119877119903119886119889 119901119897119886119905119890 (28)



119877V119894119904119888 = radic 1205881205831198861198941199031205962 (29)


119877119903119886119889 119886119894119903 119901119894119904119905119900119899 = 119899ℎ119900119897119890120588119888 (1205871198862) (1 minus 21198691 (2120596119886119888)2120596119886119888 ) (30)


119877119886 = 119899ℎ119900119897119890 (2120587119886119905 + 41205871198862) 119877V119894119904119888 + 119877119903119886119889 119886119894119903 119901119894119904119905119900119899 (31)


119880 = 119860119880119901 + 119878119880119886 (32)


119901 = 1198941205961205881198802120587119889 (33)



Re119898119900119889119894119891119894119890119889 = 120596120588ℎ2120583119886119894119903 ≪ 1 (34)









Cp(T[1K])








3 Results


(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity











(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




120574119886 = 119877119886119898119886 120574119901 = 119877119901119898119901

120596119901ℎ4 = 120596ℎ212059611988621205961198862 = 1205831198602119898119901

(13)






119860 = 1205871198772 minus 119878 (16)

119881 = 1205871198772ℎ (17)

119898119901 = 18119860119905120588119898 + 119898119903119901 (18)


119898119903119901 = minus120588119888119860120596 [[201198652 (2 (120596119888) radic119860120587)




119898119903119886 = 119899ℎ119900119897119890120588119888 (1205871198862)120596 [21198671 (2 (120596119888) 119886)(2 (120596119888) 119886) ] (21)


1198650 = 10038161003816100381610038161198651198861198881003816100381610038161003816 = 1205760119860119881119889119888119881119886119888ℎ2 (22)



120590119904119902119890119890119911119890 = 121205831198861198941199031205961198772119875119886ℎ2 (23)


119877119904119902119890119890119911119890 = 31205871205831198861198941199031198774120581 (120573)2ℎ3119899ℎ119900119897119890 (24)

where


120573 = 119886(1051199031radic2) (26)





(27)



119877119901 = 119877119904119902119906119890119890119911119890 + 119877119903119886119889 119901119897119886119905119890 (28)



119877V119894119904119888 = radic 1205881205831198861198941199031205962 (29)


119877119903119886119889 119886119894119903 119901119894119904119905119900119899 = 119899ℎ119900119897119890120588119888 (1205871198862) (1 minus 21198691 (2120596119886119888)2120596119886119888 ) (30)


119877119886 = 119899ℎ119900119897119890 (2120587119886119905 + 41205871198862) 119877V119894119904119888 + 119877119903119886119889 119886119894119903 119901119894119904119905119900119899 (31)


119880 = 119860119880119901 + 119878119880119886 (32)


119901 = 1198941205961205881198802120587119889 (33)



Re119898119900119889119894119891119894119890119889 = 120596120588ℎ2120583119886119894119903 ≪ 1 (34)









Cp(T[1K])








3 Results


(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity











(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119877119901 = 119877119904119902119906119890119890119911119890 + 119877119903119886119889 119901119897119886119905119890 (28)



119877V119894119904119888 = radic 1205881205831198861198941199031205962 (29)


119877119903119886119889 119886119894119903 119901119894119904119905119900119899 = 119899ℎ119900119897119890120588119888 (1205871198862) (1 minus 21198691 (2120596119886119888)2120596119886119888 ) (30)


119877119886 = 119899ℎ119900119897119890 (2120587119886119905 + 41205871198862) 119877V119894119904119888 + 119877119903119886119889 119886119894119903 119901119894119904119905119900119899 (31)


119880 = 119860119880119901 + 119878119880119886 (32)


119901 = 1198941205961205881198802120587119889 (33)



Re119898119900119889119894119891119894119890119889 = 120596120588ℎ2120583119886119894119903 ≪ 1 (34)









Cp(T[1K])








3 Results


(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity











(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Cp(T[1K])








3 Results


(a)

(b)

Helmholtz hole

Support

Membrane

Cavity

Insulator layer

PML

Air domainCMUT

cell

(c)

Membrane

Cavity











(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








(35)

where

119902119886 = (2120587119891119901119888 )2 ( 119881119899ℎ119900119897119890120587) (36)

119902119887 = 06radic120587119899ℎ119900119897119890119877 (37)










01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



01

1

10

Double caseM

agni

tude

of c

ompl

ex am

plitu

deof

air p

iston

velo

city

|U

a| [m

s]

frequency (kHz)



30 40 50 60 70 80 90 100 110 120 130 140 150 160

(a)

001

01

1Double case

Mag

nitu

de o

f com

plex

ampl

itude

of m

embr

ane v

eloci

ty |

Up|

[ms

]

frequency (kHz)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

|Up||Up|

(b)


30 40 50 60 70 80 90 100 110 120 130 140 150 160

1E-8

1E-7

1E-6

1E-5 Double case

Mag

nitu

de o

f com

plex

ampl

itude

of

frequency (kHz)

tota

l vol

ume v

eloci

ty |

U|[Ｇ

3s

]

FEM a=40um |U||U|

|U|









0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case|S

Ua|

|AU

p||U

|

frequency (kHz)

of a=

40 u

m d

imen

sion

[Ｇ3s

]

f- f+

fpfℎ


|SUa||AUp||U|

FEM no hole|U|

(a)

0 20 40 60 80 100 120 140 160 180 2001E-9

1E-8

1E-7

1E-6

1E-5 Double case

|SU

a||A

Up|

|U|

frequency (kHz)

of a=

160

um d

imen

sion

[Ｇ3s

]

f- f+

fp fℎ


|SUa||AUp||U|

FEM no hole|U|

(b)




30 40 50 60 70 80 90 100 110 120 130 140 150 1609095

100105110115120125130135140145150155160 Double case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)




95100105110115120125130135140145150155

Standard case

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)

frequency (kHz)


80 100 120 140 160 180 200 220 240 260 280 300 320







150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





150 200 250 300 350 400 450 500 550 60095

100105110115120125130135140145150155

Half case

frequency (kHz)

Soun

d pr

essu

re le

vel (

SPL)

ove

r th

e mem

bran

e and

hol

es (d

B)













0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)







0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

Dou

ble c

ase

SPL

at d

=360

0um

poi

nt (d

B)

frequency (kHz)

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)0 50 100 150 200 250 300 350 400

0

20

40

60

80

100

120

140

Stan

dard

case

SPL

at d

=180

0um

poi

nt (d

B)

frequency (kHz)

0 100 200 300 400 500 600 700 8000

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)0 100 200 300 400 500 600 700 800

0

20

40

60

80

100

120

140

Hal

f cas

eSP

L at

d=9

00um

poi

nt (d

B)

frequency (kHz)



4 Discussion





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Eq(6)

Eq(39)

defined step size

Input h

Eq(35) to (37)(17)



Input a

Eq(15) to (22)Eq(28) (31)

Eq(12) (13)

Eq(11a) (11b)

Calculate U

Eq(32)

Eq(33)

Calculate p

No

Yes


accurate results

Set Solve


No

Yes








[05 15

Calculate p

Calculate isc



[4isc

ℎ=p

aini

ainiaini

ainiaini

Calculate Up Ua

Calculate a p

SPL=20ＦＩＡ10(p


S A mp ma F0 Rp Ra

ℎmax]

a2pℎ

4D
















0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia










0 100 200 300 4005060708090

100110120130140150160




frequency (kHz)


the m

embr

ane a

nd h

oles

(dB)

Soun

d pr

essu

re le

vel (

SPL)

ove

r


0 200 400 600 8005060708090

100110120130140150160




frequency (kHz)


Soun

d pr

essu

re le

vel (

SPL)

ove

rth

e mem

bran

e and

hol

es (d

B)Half case





120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



120 140 160 180 200 220 240105

110

115

120

125

130

135

140 Standard case

SPL

over

the m

embr

ane a

nd h

oles

(dB)

frequency (kHz)





5 Conclusions


Data Availability




Acknowledgments




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




References








































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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