Modelling measurement microphones using BEM with visco-thermal losses

Transcription

1 Modelling mesurement micropones using BEM wit isco-terml losses Vicente Cutnd Henríquez nd Peter M. Jul Institute of Tecnology nd Innotion, Uniersity of Soutern Denmrk Cmpusej 55, 530 Odense M, Denmrk For mny decdes, models tt cn explin te beiour of mesurement condenser micropones e been proposed in te literture. Tese deices e n pprently simple working principle, crged cpcitor wose crge ries wen one of its electrodes, te diprgm, moes s result of sound wes. Howeer, mesurement micropones must be mnufctured ery crefully due to teir sensitiity to smll cnges of teir pysicl prmeters. Tere re different elements in micropone, te diprgm, te gp beind it, bck city, ent for pressure equliztion nd n externl medium. All tese subsystems form strongly coupled deice tt cnnot be modelled properly s superposition of submodels, but rter s wole. For tis reson, te cllenge of micropone modelling is still n ongoing re of reserc. In tis work, newly deeloped Boundry Element Metod implementtion tt includes isco-terml losses is used to model mesurement condenser micropones. Te models presented re fully coupled nd include FEM model of te diprgm. Te beiour of te coustic ribles in te gp nd te effect of te pressure equliztion ent re discussed, s well s te prcticl difficulty due to te production ribility mong single units of te sme micropone model. 1 Introduction Mesurement of sound pressure is mostly performed using micropones of te condenser type. Tere re mny ritions mong tese deices, but tey ll sre common fetures. Tere is conductie diprgm, exposed to te sound wes on te one side, nd closely bcked by conductie plte on te oter side. Te two elements form cpcitor tt must be crged electriclly to crete oltge. Te rition of te cpcitnce due to te diprgm moements induced by te sound field cn ten be sensed s oltge cnges. 1 Condenser micropones e nrrow spce between te diprgm nd te bck plte, usully only few micrometers tick. Tis is desired for two resons: i) te cpcitnce is lrger te closer te electrodes, nd ii) te film of ir trpped in between cn be djusted to dmp te diprgm nturl resonnces by mens of te iscous nd terml losses. Te bck plte distnce nd design cn be cosen in order to fulfil tese objecties nd obtin te desired sensitiity, noise floor nd bndwidt of te deice. Te bck plte is in mny cses proided wit oles or slits nd connected bck city in te interior of te micropone, ll of tem crefully blnced. Figure 1 sows potogrp nd sketc of condenser micropone. Te cousticl nd mecnicl coupling of te externl medium, te diprgm, te nrrow gp of ir, te oles nd te bck city is indeed ery strong nd sensitie to prmeters suc s diprgm tension nd tickness of te ir gp. Single units of te sme mesurement micropone model differ sligtly in teir performnce nd must be djusted nd clibrted indiidully. Te design nd nlysis of condenser micropones is still minly bsed on lumped prmeter models. 3 For seerl decdes, modelling of condenser micropones s been recurring topic in te literture. 4-9 Te proposed models combine nlyticl nd lumped prmeter representtions of te elements of te deice: externl medium, diprgm, tin ir film, oles nd bck plte. Some utors introduce numericl models of some of tese elements, but in most cses te isco-terml effects in te tin ir gp re represented by nlyticl formuls or lumped prmeters. In

2 reference [9], te Finite Element Metod is used wen ccounting for iscous nd terml losses; oweer, te comprison wit mesured results of te sensitiity nd membrne displcement does not seem totlly stisfying. Te Finite Element Metod (FEM) pproc s demonstrted its lidity for simplified micropone geometry, te sme used in te first exmple in tis pper. 10,11 Viscous nd terml losses re usully neglected wen modeling sound wes, or rter ccounted for s boundry impednce. Tis is due to te fct tt most of te iscous nd terml effects occur witin tin lyer oer te boundry, wit tickness of frction of millimeter to few micrometers depending on te frequency. Howeer, setups were te dimensions re smll enoug to be of te sme rnge s te isco-terml boundry lyer need more dnced models tt include losses. To do so, linerized ersion of te Nier-Stokes equtions for fluids must be used, eiter directly in FEM implementtion or troug te Kircoff decomposition of tese equtions into coustic, terml nd iscous components In tis pper te Kircoff decomposition in combintion wit te Boundry Element Metod (BEM) is used µm Figure 1. Potogrp (source: Brüel & Kjær) nd digrm of condenser micropone, sowing its interior. Te pper will present te dpttion of custom-mde implementtion of te Boundry Element Metod, te OpemBEM, to include iscous nd terml losses. An xisymmetricl implementtion is cosen wic is conenient for te geometry of condenser micropones. Two clcultion exmples re clculted: i) n idelized geometry wit no bck city, nd ii) Brüel & Kjær mesurement micropone wit bck city nd no oles in te bck plte. Te results re compred wit n nlyticl model nd electrosttic ctutor mesurements respectiely. Visco-terml equtions Sound propgtion in fluid wit losses cn be described using tree decoupled modes: i) n coustic mode describing te prt of te we tt propgtes, ii) terml mode, wic includes te effect of terml losses nd iii) iscous mode, wit te losses by iscous friction Te rmonic equtions representing te tree effects re: ( + k ) p = 0 ( + k ) p = 0 ( + k ) = 0 wit = 0 (1,,3) were te indexes, nd indicte respectiely coustic, terml nd iscous modes nd te totl pressure is p = p + p, since te iscous elocity s no pressure ssocited. Te wenumbers k, k nd k re function of te perfect fluid wenumber k nd seerl pysicl prmeters, suc s iscosity, bulk iscosity, terml conductiity, etc. Te rmonic time dependence e iωt is omitted. Equtions (1), () nd (3) nd te modes tey represent re decoupled in te fluid, but tey re linked troug te boundry conditions. Te iscous nd terml modes re prticulrly strong witin tin boundry lyer on te boundry surfce, cused respectiely by te no-slip condition of te tngentil elocity nd te terml bsorption of te surfce. Te iscous nd terml boundry lyers e tickness tt is inersely proportionl to te squre root of te frequency, nd t udio frequencies it spns from frction of millimeter to few micrometers. Te boundry condition for te pressure is consequence of te ig terml conductiity te boundry, creting isoterml conditions. Since te pressure is relted to te temperture, it must be:

3 T = T + T = τ p + τ p = 0 (4) Te prmeters τ nd τ depend on te pysicl properties of te fluid nd te frequency. Te elocity boundry condition, in opposition to te clssicl norml elocity condition for perfect fluids, s ectoril nture for fluids wit losses. Tis mens tt te totl elocity ector, te sum of ll tree modes, must mtc te moement of te boundry in te norml (noted n) nd tngentil (noted t) directions, tt is, = + +. boundry If te terml nd coustic elocities re expressed s function of te corresponding pressures: = φ p + φ p + (5) boundry were φ nd φ depend on pysicl constnts nd te frequency. Tis ectoril condition is split into norml nd tngentil components, te ltter being two-dimensionl ector expression: boundry, n = + φ +, n p p φ (6) n n = φ p + φ p + (7) boundry, t t t, t Eqution (7) is te so-clled no-slip condition becuse te fluid must follow te tngentil moement of te setup t te boundry. 3 Te Boundry Element Metod wit isco-terml losses. 3.1 Axisymmetricl BEM Equtions (1) to (3) re ll of te Helmoltz type (note tt Eq. (3) is ectoril nd in relity represent tree equtions one for ec component) nd terefore well suited for te Boundry Element Metod. Howeer, te numericl implementtion s certin differences wen compred to implementtions or te stndrd (lossless) Helmoltz eqution, since te wenumbers k nd k e lrge imginry lues nd represent fst decying fields, wic must be crefully ndled in te numericl integrtion. It is beyond te scope of te present pper to gie detiled ccount of te mtemticl mnipultions, but using te boundry conditions (Eq s (4), (6) nd (7)) llows for te elimintion of ribles nd results in n expression relting te coustic pressure p to te norml component of te boundry elocity boundry,n ssuming tt te tngentil boundry elocity is zero: = VTp (8) boundry,n were te mtrix VT is combintion of mtrices resulting from te discretized ersions of Eq. s (1) to (3) linked toug te boundry conditions In tis pper n xisymmetricl formultion of te BEM is used, wic s been progrmmed in te OpenBEM Mtlb code. Te objects to be modeled must e xil symmetry, nd only teir genertor needs to be mesed. 4 Model of n idelized micropone A simple geometry of micropone wit n nlyticl solution cn be found in reference [11]. Tis exmple s been used preiously in te literture for comprison wit numericl models. 10,14 Te micropone is circulr diprgm stretced oer circulr city nd rd bck plte, forming flt cylinder. Te rim of te structure is set wit pressure relese boundry condition (p=0) nd excited by uniform sound pressure on te membrne. Te setup is sketced in figure. Te surfces of diprgm nd bck plte re close enoug to cuse problems in stndrd BEM formultion; tis difficulty s been delt wit using n improed integrtion strtegy. 19 In principle, te boundry surfce must be differentible t lest once (no srp corners) in order for te surfce derities to exist. 13 Srp corners re neerteless implemented in order to obsere te effect on te solution.

4 Figure. Sketc of te idelized micropone in reference [11]. Te rdius of te micropone is mm nd te gp tickness is 18 µm. Te diprgm tension is 318 N/m, te density of te diprgm mteril is 8300 kg/m 3 nd te diprgm tickness is 6,95 µm. 4.1 Model of te diprgm Te diprgm is modelled s membrne wit no stiffness, following te eqution: were ε is te norml displcement, K is te wenumber of te mecnicl we, T is te membrne tension nd p d is te sum of sound pressures cting on te diprgm, internl nd externl. 4,5 Tis eqution is implemented using onedimensionl Finite Element Metod implementtion producing te mtrix eqution: 4. Coupled model of te idelized micropone Te coupled system of equtions for te idelized micropone is: T( A + K B) iω were eqution (10) cn be seen in te top rows of te system of equtions. Te term VT represents eqution (8) nd D nd iω re coupling mtrices: D represents forces on te diprgm from te coustic c pressure nd iω represents te membrne moement s boundry elocity for te coustic prt. Te p re te pressures on te nodes in te interior of te micropone, p inc re te contributions of te externl pressure on te diprgm nodes, nd te ε i re te displcements of te diprgm nodes. 4.3 Results r ε 1 ε + + K r r ( A K B ) + ε Te sensitiity of te idelized micropone is clculted by integrting te diprgm displcement oer te re of te diprgm wic is proportionl to te sensitiity of te micropone 14,17. Te sensitiity sown in figure 3 s been clculted for rnge of meses wit rying number of elements nd llowing in some cses srp corners on te rim. Te clcultion does not brek down wen te mes density is reduced or te corners re mde srp, nd te greement for te rouger meses cn be cceptble for initil clcultions. Te effect of rounding te corners is less eident te more elements in te mes. Te finer mes, wit 176 elements, is lmost identicl to te nlyticl solution. All cures, nlyticl nd numericl, re normlized wit te sme lue. Bot nlyticl nd numericl solutions re ble of producing results of te iscous, terml nd coustic mgnitudes independently, bot on te boundry nd in te micropone s interior. As n exmple, te rdil component of te prticle elocity is sown in figure 4 for seerl frequencies long line in te z direction nd t distnce midwy ε = m m = p T p d T ε i p D = VT p 0 d inc (9) (10) (11)

5 from te centre to te rim. Te boundry lyer cn be seen clerly: te fluid is forced to be still by te diprgm nd bck plte surfces. Figure 5 sows membrne displcement for te sme frequencies nd long rdius. Te mtc wit te nlyticl solution is quite good in ll cses, so muc tt te cures re lmost identicl. Te results sown in figures 3, 4 nd 5 were clculted wit one of te finer meses, wit 176 elements. Te lues sown in figure 4 re clculted in te domin, tt is, te BEM results on te surfce re used in n dditionl clcultion were domin points re obtined. Te process is not direct in te cse of te iscous component of te prticle elocity, since te surfce results re gien in boundry reference system (tngentil nd norml components), wic needs to be trnsported to te globl coordinte system. On te oter nd, te terml nd coustic components of te prticle elocity re clculted s finite difference grdients of te domin pressures. Te outcome is oweer stisfctory. Figure 3. Normlized sensitiity of te idelized micropone. Te lower plot is zoomed-in ersion of te upper one. Continuous blck line, nlyticl solution; x, 31 elements, rounded corners; o, 5 elements, srp corners; +, 45 elements, srp corners; *, 51 elements, rounded corners;, 100 elements, rounded corners;, 176 elements, rounded corners.

6 Figure 4. Rdil component of te prticle elocity long erticl line trersing te gp lfwy from te centre to te rim. *, BEM clcultion; continuous line, nlyticl solution. Figure 5. Membrne displcement, oer rdil direction. *, BEM clcultion; continuous line, nlyticl solution. 5 Model of mesurement condenser micropone Te second exmple is commercil mesurement condenser micropone, were te coupled model clcultions re compred wit mesurement results. Te mesured sensitiity frequency response is tken from te literture nd is n electrosttic ctutor mesurement 14. Te cosen micropone is te Brüel & Kjær micropone type 4938, ¼ʺ pressure-field micropone wit no oles in te bck plte. 0 Te micropone is terefore xisymmetricl nd is suitble for modelling using te xisymmetricl BEM. Te only non-xisymmetricl feture is te pressure equliztion ent on te side, wic is modelled s pressure-relese circulr slit. Te nominl prmeters of te 4938 re listed in Tble 1. Howeer, during production, te micropone is indiidully djusted in order to meet te model specifictions. As result, te micropone prmeters nd in prticulr te gp widt nd diprgm tension ry from unit to unit nd must be djusted in te numericl model. Figure 6 sows picture of te 4938 nd exmples of BEM meses employed to model te micropone s internl nd externl boundries. Note te different scles of te sub figures. Tble 1. Design prmeters of te Brüel & Kjær micropone type Membrne rdius (mm) Bckplte rdius (mm) 1,75 Gp tickness (µm) 18 Membrne tension (N/m) 318 Membrne density (kg/m 3 ) 8300 Membrne tickness (µm) 6,95

7 Figure 6.From left to rigt, picture of te type 4938 micropone nd exmples of BEM genertor s meses of te interior nd exterior domins. 5.1 Coupled model of te idelized micropone Te coupled system of equtions for te BK4938 micropone is s follows: Eqution (1) includes te sme terms s eqution (11), wit te dditionl coupling of te externl domin, represented by te BEM coefficient mtrices A ext nd B ext. Te term VT represents eqution (8), te term T(A+K B) is eqution (10), nd D, E nd iω re coupling mtrices. Te p re te pressures on te nodes in te interior of te micropone, te p e re te scttering pressures on te externl surfce of te micropone nd p inc re te contributions of te externl incident sound field on te externl surfce. All te essentil elements tt ply role in te performnce of te micropone re represented in tis coupled system, wit te exception of te electrosttic force. Tis force is not ery relent for tis micropone type ccording to personl communiction wit Brüel & Kjær, nd te results in tis pper seem to support te ide. Electrosttic forces cn esily be dded s n excittion to te diprgm nodes witout extending te system of equtions. 5. Results T(AA + K B) iω iω D VT 0 B ε i 0 = p 0 p e p Figure 7 compres te mesured responses nd te corresponding djusted models. In figure 7, te model prmeters re : gp tickness of 19,65 µm (+9%), diprgm tension of 3566 N/m (+14%), nd diprgm surfce density of 61,1 g/m (+6%). Numbers in prenteses re reltie deitions from te nominl lues E 0 1 ext A ext inc (1)

8 Figure 7. Sensitiity response: comprison between te ctutor mesurement in reference [14] nd te djusted numericl model. x, BEM clcultion, continuous line, ctutor response. 6 Conclusions Te min purpose of tis pper, te modelling of te mesurement condenser micropone, is cieed by mens of noel xisymmetricl BEM formultion wit losses of te interior of te deice tt is included in coupled model. Te condenser micropone poses prticulrly cllenging modelling tsk, nd it cts terefore s test for te new BEM implementtion. Besides te inerent complicted nture of condenser micropones, teir modelling is mde yet more difficult becuse tey re djusted one by one during production in wy tt preents sufficiently precise knowledge of teir key prmeters. Bot mesurements nd simultions demonstrte tt quite smll cnges in te construction prmeters of te micropone cn gie rise to significnt cnges in its performnce. Te simultions of teoreticl micropone in section 4 demonstrte tt te numericl model is ble of coping wit te tsk. Wen rel micropone is simulted in section 5, oweer, prmeter djustment process is necessry in order to get close model, nd een ten it is not possible to ssume tt te cosen set of prmeters reflects relity. Tis is due to te micropone complexity, not result of ny modelling difficulty. Een in te cse of te rel micropone in section 5 nd te fitting procedure, te sensitiity obtined differs only by frction of 1 db. Te proposed model cn neerteless be used for studying te pysicl beiour of te micropone, improe its design nd dpt it to te systems were it is included. Primry nd secondry clibrtion systems, couplers, ering ids nd mobile deices cn benefit from n improed micropone model. As to te BEM model wit losses, it s sown its dequcy for tis cllenging modelling tsk wit meses tt e fewer degrees of freedom tn te corresponding FEM models in te literture. Te model is robust enoug to cope wit srp edges nd corners, wic in principle sould be rounded for precise representtion. Te use of te Kircoff decoupling of iscous, terml nd coustic modes combines well wit te Boundry Element Metod, since ll te coupling is performed t te boundry, were te BEM boundry conditions re pplied. If te coefficients of te BEM mtrices re clculted properly, clcultion in te domin follows esily.

9 7 Acknowledgement Te utors wis to tnk Erling Sndermnn Olsen nd Jon Grmtorp from Brüel & Kjær, for teir interest nd support in tis project. References [1] L. L. Bernek, Acoustics. Acousticl Society of Americ, New York, [] Brüel & Kjær Micropone Hndbook. Brüel & Kjær Sound & Vibrtion Mesurement A/S, Nærum, Denmrk, [3] F. V. Hunt, Electrocoustics. Te Anlysis of Trnsduction, te nlysis of trnsduction nd Its Historicl Bckground. Acousticl Society of Americ, New York, 198. [4] D. H. Robey, Teory of te Effect of Tin Air Film on te Vibrtions of Stretced Circulr Membrne. Journl of te Acousticl Society of Americ, 6(5), , [5] A. J. Zuckerwr, Teoreticl response of condenser micropones. Journl of te Acousticl Society of Americ, 64(5), , [6] X. Bo nd Y. Kgw, A Simultion of Condenser Micropones in Free Field by Boundry Element Approc. Journl of Sound nd Vibrtion 119(), , [7] R. S. Grinnip III, Adnced Simultion of Condenser Micropone Cpsule. J. Audio Eng. Soc., Vol. 54, No. 3, 006. [8] Toms Lergne, Stépne Durnd, Micel Bruneu, nd Nicols Joly, Dynmic beior of te circulr membrne of n electrosttic micropone: Effect of oles in te bcking electrode. Journl of te Acousticl Society of Americ 18 (6), , 010. [9] D. Homentcosci nd R. N. Miles, An nlyticl-numericl metod for determining te mecnicl response of condenser micropone, Journl of te Acousticl Society of Americ 130 (6), , 011. [10] W. R. Kmping, Y. H. Wijnnt nd A. de Boer, An Efficient Finite Element Model for Viscoterml Acoustics, Act Acustic united wit Acustic, 97, , 011. [11] G. Plntier nd M. Bruneu, Het conduction effects on te coustic response of membrne seprted by ery tin ir film from bcking electrode. Acoustique 3, 43-50, [1] A. D. Pierce, Acoustics. An Introduction to Its Pysicl Principles nd Applictions. Acousticl Society of Americ, New York, [13] M. Bruneu, P. Herzog, J. Kergomrd nd J. D. Polck: Generl formultion of te dispersion eqution in bounded isco-terml fluid, nd ppliction to some simple geometries. We Motion Nort Hollnd 11, , 1989 [14] V. Cutnd Henríquez, Numericl trnsducer modeling. PD tesis, Tecnicl Uniersity of Denmrk, Lyngby, Denmrk, 00. [15] V. Cutnd Henriquez, P. M. Jul : OpenBEM - An open source Boundry Element Metod softwre in Acoustics. Proceedings of Internoise 010. October 010, Lisbon. [16] Vicente Cutnd Henríquez nd Peter Møller Jul, An Axisymmetricl Acoustic BEM Formultion Including Visco-Terml Losses. Proceedings of Forum Acusticum 011, Alborg, Denmrk, 011. [17] P. M. Jul, Te Boundry Element Metod for Sound Field Clcultions. PD Tesis. Tecnicl Uniersity of Denmrk, Lyngby, Denmrk, [18] P.M. Jul, An xisymmetric integrl eqution formultion for free spce non-xisymmetric rdition nd scttering of known incident we. Journl of Sound nd Vibrtion 163 (1993), [19] V. Cutnd Henríquez, P. M. Jul nd F. Jcobsen: On te Modeling of Nrrow Gps Using te Stndrd Boundry Element Metod. Journl of te Acousticl Society of Americ, 109, (001). [0] Brüel & Kjær Product Dt Seet BP1844, ¼ʺ Pressure-field Micropone Type Brüel & Kjær Sound nd Vibrtion Mesurement A/S, Næerum, Denmrk, 008.