Displacement field of the membrane of condenser microphones at high frequencies: Improvement of analytical solution

Transcription

1 Displacement field of the membane of condense micophones at high fequencies: Impovement of analytical solution Thomas Lavegne, Stéphane Duand, Michel Buneau, Nicolas Joly To cite this vesion: Thomas Lavegne, Stéphane Duand, Michel Buneau, Nicolas Joly. Displacement field of the membane of condense micophones at high fequencies: Impovement of analytical solution. Société Fançaise d Acoustique. Acoustics 2012, Ap 2012, Nantes, Fance <hal > HAL Id: hal Submitted on 23 Ap 2012 HAL is a multi-disciplinay open access achive fo the deposit and dissemination of scientific eseach documents, whethe they ae published o not. The documents may come fom teaching and eseach institutions in Fance o aboad, o fom public o pivate eseach centes. L achive ouvete pluidisciplinaie HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau echeche, publiés ou non, émanant des établissements d enseignement et de echeche fançais ou étanges, des laboatoies publics ou pivés.

2 Displacement field of the membane of condense micophones at high fequencies: Impovement of analytical solution T. Lavegne, S. Duand, M. Buneau and N. Joly Laboatoie d acoustique de l univesité du Maine, Bât. IAM - UFR Sciences Avenue Olivie Messiaen Le Mans Cedex 9 thomas.lavegne@univ-lemans.f 3317

3 Condense micophones ae ecipocal tansduces whose popeties (sensitivity, bandwidth and eliability) make them poweful measuement tools. Fo thei common use unde standad conditions they have been appopiately chaacteised fo nealy thity yeas. But nowadays, thei miniatuisation (using MEMS pocesses) and thei new uses fo metological puposes unde non-standad conditions (i.e. in high fequency anges, in gas mixtues, and at vaious static pessues and tempeatues) equie a much deepe chaacteisation with espect to these new uses. Though ecent liteatue on this topic [J. Acoust. Soc. Am. 128(6), (2010)] leads to satisfying esults accoding to these new equiements, the analytical solution given is not always sufficiently pecise to intepet phenomena and must be impoved to chaacteise moe accuately the displacement field of the membane up to high fequencies (100 khz). Thus, the aim of the wok pesented hee is to popose an impoved analytical solution (especially by taking into account moe appopiately the finite suface coesponding to the holes in the backing electode) and to deduce esults of inteest, in paticula concening metology puposes. 1 Intoduction Condense micophones have been appopiately chaacteised fo decades fo thei common use unde standad conditions. A much deepe chaacteisation as the one available peviously is today equied due to thei the miniatuisation (using MEMS pocesses) and to thei new uses fo metological puposes unde non-standad conditions (i.e. in high fequency anges, in gas mixtues, and at vaious static pessues and tempeatues). An analytical modeling, pesented ecently in a pape [1] stating up again with a pevious one [2], which descibes the behaviou of the diaphagm of electostatic tansduces, appeas to be a convenient tool to analyse (moe paticulaly) the effect of the holes in the backing electode on the displacement field which can be highly non-unifom in the highest fequency ange. Being concened by the use of this analytical appoach when low uncetainties on the behaviou of acoustic fields geneated o measued by these tansduces ae equied up to 100k Hz, accuate theoetical esults could be obtained povided that the main geometical paametes (namely hee those chaacteising the ai gap and the holes in the backing electode) ae accounted fo in a ealistic manne. Thus, this pape aims at poviding esults fom impovements in intoducing in the calculation a moe ealistic shape to descibe the geomety of each hole, even though this shape emains educed to seveal punctual equivalent volume velocity souces. Moeove, an extended shape which accounts fo the flow though the holes and in the ai gap is intoduced, showing that accuate esults can be obtained (hee the displacement of the membane and the sensitivity), which ae moe ealistic than those pesented in the pape mentioned above. Futhemoe, numeical simulations ae concened with examining the behaviou of the acoustic velocity fields aound the inteface between the holes and the ai gap in ode to help intepeting the value of the paametes used in the analytic simulations, and expeimental esults, obtained using a lase scanning vibomete, ae pesented and compaed to the analytical esults. 2 Basic fomulation of the analytical modeling The pupose of this section is to esume the basis of the modeling pesented in the liteatue [1] in ode to emind of how the pesence of the holes and the peipheal slit ae taken into account in the basic equations which goven the coupled fields which take place in each pat of the micophone (membane, ai-gap, backchambe, holes, and peipheal slit). p av z p(, θ) 0 ξ(, θ) U(θ) U1,ν0 p C V c a Hole Peipheal slit 1 2 = a θ U(θ) U 1,ν0 2π n0 Figue 1: Diagam of a condense micophone with 6 holes in the backing electode. The displacement field ξ (,θ) of the membane (see Figue 1), diven by a hamonic acoustic pessue p av unifom ove the suface (πa 2 ) and loaded by the pessue p (,θ) in the ai-gap, which is subjected to the Diichlet bounday condition at its peiphey ( = a), is govened by the following set of equations T ( ) 2 2θθ + K2 ξ (,θ) = p av p (,θ), (1a) ξ ( = a,θ) = 0, (1b) whee K = ω M s /T, and whee ω, M s and T ae, espectively, the angula fequency, the mass pe unit aea, and the mechanical tension of the membane. In the ai-gap between the membane and the backing electode, the pessue vaiation p (,θ), which is assumed to be independent of the coodinate z (nomal to the membane), is govened by the following equation 3318

4 Figue 2: Displacement field of the membane of a 1 /2 in. B&K micophone type 4134 at 72 khz. ( ) 2 2θθ + χ2 p (,θ) = ρ 0 ω 2 i ωρ 0 F v + U (θ) n 0 ν 0 =1 U 1,ν0 (,θ) δ ( 2 ) F v δ ( 1 ) ξ (,θ) δ ( θ θ ν0 ) ], (2a) with the Neumann bounday condition at the peiphey (the adial component of the paticle velocity vanishes a) p ( = a,θ) = 0, (2b) whee is the thickness of the ai gap, ρ 0 the density of ai, n 0 the numbe of holes in the backing electode, δ is the Diac function, 1 and 2 the distance between the cente of the membane and, espectively, the cente of each hole and the cente of the peipheal slit, and whee U 1,ν0 and U (θ) ae the volume velocity of, espectively, the hole numbeed μ 0 and the peipheal slit. The complex wavenumbe χ and the function F v account fo the viscous and themal bounday layes effects, assuming the no-slip condition and the isothemal bounday condition on the membane and an appoximate mixed bounday condition on the suface of the backing electode involving its poosity (see details in [1, 3]). The fist tem in the ight hand side of Eq. 2a epesents the volume velocity of the diaphagm and shows the coupling between the displacement field ξ (,θ) and the pessue field p (,θ), and the second tem epesents the volume velocities (localized sinks) along the z-axis of each hole U 1,ν0 and of the peipheal slit U (θ). These volume velocities souces, which ae assumed to be punctual (that is dastic an appoximation), can be expessed as a function of the pessue vaiation in the ai gap and the pessue vaiation in the backchambe as follows [ U 1,ν0 = y 1,ν0 pc ( 1,θ= 0)) p( 1,θ= 0) ], (3a) U (θ) = y [ p C ( 2,θ) p( 2,θ) ], (3b) whee y 1,ν0 and y ae the input admittance of each hole and of the peipheal slit espectively. In the backchambe, the acoustic pessue field p C (,θ), which is subjected to Neumann bounday condition at the peiphey ( = a), is govened by the following set of equations ( ) 2 2 θθ + χ2 C p C (,θ) = i ωρ 0 n 0 U 1,ν0 (,θ) δ ( 1 ) F vc δ ( ) θ θ ν0 ν 0 =1 C + U (θ) ] δ ( 2 ), (4a) C p C ( = a,θ) = 0, (4b) whee C is the aveage thickness of the backchambe, and whee χ C and F vc take into account the viscous and themal bounday layes effects involving the poosity of the backing electode. This fomulation leads to solutions, involving modal expansions on Diichlet and Neumann eigenfunctions, which depend on the adial and the azimuthal coodinates, suitable to descibe the inticate behaviou of the membane in the highest fequency ange. 3 Results and impovement of the analytical solution The theoetical displacement field of the membane shown in the left hand side of Figue 2 is obtained with the solution given in [1] and computed with the values of the geometical, mechanical and electical paametes given in [1, 2] fo a 1 /2 in. B&K micophone type The significant deviations appeaing between this theoetical esult and the expeimental one (shown in the cente of Figue 2) may be due (at least patially) to the fact that the holes and the peipheal slit ae modelled as Diac souces localised at the cente of each hole and of the peipheal slit (punctual and linea sinks espectively). This appoximation, although appopiate to descibe the inticate behaviou of the membane in the highest fequency ange, is too dastic to chaacteize accuately the behaviou of the fluid in the ai gap, and, consequently, the influence of the souces on the displacement field of the membane. This is paticulaly tue at cetain fequencies (hee 72 khz) whee the displacement field is highly non-unifom. 3319

5 Moe accuate esults can be obtained by accounting fo moe ealistically the shape of the holes and the slit. Hence, each hole and the slit ae descibed by, espectively, seveal punctual and seveal linea equivalent volume velocity souces. The theoetical displacement field of the membane shown in the ight hand side of Figue 2 is obtained when the slit is descibed by two linea souces located at the coodinates ( 2 e/2) and ( 2 )(e denoting the width of the slit), and when each hole is descibed by fou punctual souces whose locations ae on a adius of the micophone at the coodinates ( 1 ± R 1,θ ν0 ) and between two holes at the coodinates ( 1,θ ν0 ± dθ) with dθ = atan(1.7r 1 / 1 ). It appeas clealy that the theoetical displacement field of the membane (calculated using this paametes) looks simila to the measued one, and to the one simulated in [4] (note that in this efeence the displacement scale is not coect). Figue 3 shows the coss-sections of the thee displacement fields epesented on Figue 2 along a diamete passing though two holes. It is notewothy that using the analytical solution pesented hee the amplitude of the displacement is quite well evaluated. Howeve, the location of the maximum displacement does not coespond to the expeimental one which is located (appoximately) in font of the holes (i.e. fo 1 2mm) (actually this discepancy depends on the appoximated desciption of the extended souces ). ξ(,θ) (m) x 10 8 Theoetical (pevious) Expeimental Theoetical (cuent) (m) x 10 3 Figue 3: Coss-sections of the displacement fields shown on Figue 2. The locations of the punctual and linea souces used hee to descibe the holes and the slit show that they behave as extended volume velocity souces. To help us to undestand the values of the paametes used (i.e. the locations of the souces ), a numeical simulation accounting fo the viscous and heat conduction effects in the bounday layes has been pefomed with an axisymetical modelling using an anisotopic mesh [5] in ode to examine the behaviou of the acoustic velocity fields aound the inteface between the holes and the ai gap. In this axisymetical simulation the holes in the backing electode ae eplaced by an annula slit, whose suface coesponds to the total suface of the n 0 = 6 holes. The esult of the numeical simulation shown on Figue 4, obtained when consideing an unifom velocity field of the membane, epesents the diection of the paticle velocity (white aows) and the amplitude of its axial component (colo chat) at the mouth of the slit, on the side of the ai gap. It appeas clealy that the amplitude of the axial velocity is maximum at the shap edges of the slit. Thus, owing to this obsevation, it seems moe appopiate to descibe each holes by seveal punctual souces localised at the peiph- 1 R 1 1 v z v z max Figue 4: Axial component v z of the paticle velocity at 72 khz (Aows: Oientation of the paticle velocity ; Colomap: Nomalised magnitude of the axial component of the paticle velocity). ey of the holes instead of a single punctual souce located at each of thei centes. Note that the axial velocity is geate on the left side of Figue 4 (towads the cente of the membane) than on the othe side, so, in othe wods, the volume velocity souces do not have the same weight accoding to thei locations (this should lead to theoetical esults close to the expeimental one than that shown on Figues 2 and 3). Thus, to compensate the fact that the modelling [1] does not take this phenomenon into account the locations of the punctual souces would be eaanged accoding to thei diffeent weights. 4 Sensitivity of the micophone The sensitivity as a function of the fequency ( 1 /2 in. B&K micophone type 4134) obtained fom the analytical pocedue outlined in section 2, when modeling holes and peipheal slit as indicated in section 3, is shown in Figue 5 fo the configuation outlined in table 1 and fo the values of the paametes given in ef. [2], when consideing one azimuthal mode (m = 0) and sixteen adial modes (n = 0to15) in the calculus: expeimental esult (uppe cuve, black solid line with dots), pevious [1] theoetical esult (lowe cuve ed dotted line), cuent esult (intemediate blue solid line). As expected these esults show clealy seveal featues: fist, the elatively good ageement between the cuent analytical esult and the expeimental one, especially concening the vey low fequency shift between the theoetical and expeimental values of the esonance; second, the emaining discepancies between the theoetical and expeimental associated Q-facto which could be explained by the (unknown) uncetainties on the values of the geometical, mechanical and electical given in ef. [2]; thid, the systematic lowe theoetical level ( 4 db) which is cuently non explained (it is the fist time that such theoetical sensitivity cuve is

6 Modulus (db ef. to 1V/Pa) Expeimental Theoetical (pevious) Theoetical (pesent) Fequency (Hz) Figue 5: Sensitivity of the micophone vs fequency. calculated fom the geometical, mechanical and electical paametes). Table 1: Values of the modified paametes used to compute the sensitivity shown on Figue 5 and of the paametes given in [2]. Paamete Value fom [2] Value used hee Radius of a hole R 1 0.5mm R 1 0.5mm Location of the punctual souces Width of the peipheal slit Location of the linea souce 1 2mm 1 = 1 R mm a mm a mm 2 4mm 2 = 2 a 2 / mm It is woth noting that non-linea effects which could appea aound the shap edges of the holes and the slit, namely sepaation of flow and even fomation of votices, could be non negligible fo high incident pessue vaiation levels (they ae not accounted fo in the cuent modeling). Refeences [1] T. Lavegne, S. Duand, M. Buneau, D. Rodigues and N. Joly, Dynamic behavio of the cicula membane of an electostatic micophone: Effect of holes in the backing electode, J. Acoust. Soc. Am. 128(6), (2010) [2] A.J. Zuckewa, Theoetical esponse of condense micophones, J. Acoust. Soc. Am. 64(5), (1978) [3] R.S. Ginnip, Advanced Simulation of a Condense Micophone Capsule, J. Aud. Eng. Soc. 54(3), (2006) [4] D. Homentcovschi, R.N. Miles, An analyticalnumeical method fo detemining the mechanical esponse of a condense micophone, J. Acoust. Soc. Am. 130(6), (2011) [5] N. Joly, Finite Element Modeling of Themoviscous Acoustics on Adapted Anisotopic Meshes: Implementation of the Paticle Velocity and Tempeatue Vaiation Fomulation, Acta Acustica united with Acustica 96, (2010) 5 Conclusion The aim of this study was to detemine whethe paametes adjusted to model the phenomena which occu aound the holes in the ai gap significantly impove esults which can be obtained fom an analytic appoach ecently published. The esults gatheed ae consistent with this hypothesis, the conclusive numeical and expeimental esults (including those available ecently in the liteatue [3]) confiming that. Futhe eseaches will conside othe geometical, mechanical, and electical paametes and the use of the model in eal situations concening miniatuization and/o metological applications. It is notewothy that the theoetical esults convey an intepetation of the physical phenomena, and that, while the analytical modelling pesented hee could appea somewhat cumbesome, the numeical calculations ae in fact vey simple and apid to handle. 3321