Modeling acoustic transducer surface waves by Transmission Line Matrix method

Transcription

1 Modelng acoustc transducer surface waves by Transmsson Lne Matrx method Andreas Wlde Fraunhofer Insttut für Integrerte Schaltungen, Außenstelle EAS Peter-Chrstan Eccardt Semens AG, CT MS, München Wllam O Connor Unversty College Dubln Summary A model for the generaton and propagaton of acoustc and surface waves s mplemented employng the Transmsson Lne Matrx (TLM) method wth a locally reactng surface restorng boundary condton, whch s a model for an array of capactve ultrasound transducers. Frst test calculatons clearly show the ablty of the proposed model to capture acoustc and surface wave effects. The comparson wth a reference soluton derved wth FE methods reveals that further refnements may be necessary although general agreement s acheved. Keywords TLM, surface waves, ultra sound, transducer array on FEM Technology 1

2 1 Introducton Mcro-mechanc ultrasound transducer arrays are usually bult on the bass of pezo electrc bulk resonators. Alternatve transducer prncples, especally capactve transducers are not employed because bulk resonators offer hgher sound pressure levels, however wth the drawback of a small bandwdth. In the last few years technologes were developed that allow to produce capactve transducers wth extremely small gap szes (gap sze means dstance between the membrane and the counter electrode) on the bass of the standard BCMOS process. Ths small gap enables hgh sound pressure levels and a large bandwdth at reasonable load voltages (about 10 V). Arrays of capactve transducers consst of a large number of membranes next to each other whch are set n moton by an alternatng current. Experments have shown that apart from sound waves n the plane of the transducer array surface waves may be generated and propagate, much lke the gravty waves at the surface of a lake. Ths surface waves have long rng down tmes and may cause serous errors f the transducer array s used as a mcrophone. Therefor a method s needed to calculate the generaton and propagaton of these surface waves n order to develop and test strateges to suppress the unwanted surface effects. The Fnte Element Method (FEM) s a well-establshed tool for the numercal soluton of partal dfferental equatons. One dsadvantage of the method results from the fact that the numercal effort tends to grow as the square of the number of nodes whch dscretze the problem. Thus the demand of a three-dmensonal calculaton grows wth the power of sx of a typcal length. Unfortunately to resolve propagatng waves the node densty must not fall below a certan lmt whch prevents an enlargement of the element sze far from the orgn. Ths constrant makes the calculaton of larger acoustc felds mpossble. On the other hand, for wave propagaton other numercal schemes exst whch requre computatonal resources drectly proportonal to the number of nodes. In ths case the total cost of the computaton s proportonal to the power of three of a typcal length of the problem n three dmensons. One such scheme s the Transmsson Lne Matrx (TLM) method. The theoretcal foundatons of the TLM method date back to the 40 s when Whnnery et al. and Kron developed equvalent crcut models made up of dscrete elements for Maxwell s equaton to study wave gude models usng network analyzers [1,, 3]. Wth the advent of more powerful computers the TLMmethod was made popular by Johns and Beurle n 70s. Snce then the method has been appled successfully to many problems whch are descrbed by wave or dffuson equatons. The man dfference between the TLM ansatz and standard schemes such as Fnte Dfference or FE methods s as follows: Usually a physcal problem s descrbed precsely by a partal dfferental equaton whch subsequently s dscretzed. The soluton to ths equaton s then approxmated numercally. The TLM approach s to approxmate the physcal process (e.g. wave propagaton) by dscrete physcal models, whch can then be calculated exactly. Ths approach has several advantages. The physcal models may be programmed ntutvely, all dscretzaton approxmatons are apparent, and the calculaton may be parallelzed easly by doman decomposton. The dsadvantages are that TLM s best suted to equdstant nodes and fxed tme steps, whch are coupled to the spatal dstance of the nodes. However, n recent years TLM was adapted for non-equdstant and curvlnear grds. For complex techncal devces such as acoustc transducer arrays t s requred to compute several coupled physcal effects (electro-magnetcs, structural mechancs, flud structure nteracton) whle large sound felds must be mapped. One approach s to couple TLM and FEM usng FEM to calculate the structural and electromagnetc effects and usng TLM to model the acoustc flud. For ths study a major problem was to smulate surface waves whch are generated at the boundares of the actve area of a transducer array and whch then propagate along the surface of the transducer array. These surface waves decay slowly and may cause sgnfcant deflectons of the transducer membranes whch are not wanted as they are not drectly related to the acoustc feld. As a frst step the ablty of the TLM-model to on FEM Technology

3 smulate surfaces waves on a transducer array coupled to a flud s studed. The next secton descrbes the TLM theory and algorthm n more detal. Secton 3 deals wth the couplng of TLM and a smple model for the transducer array, whch s to be replaced by FEM models n future. In the followng secton the current status of the work s reported by gvng an example of a flud/structure couplng. TLM-Theory The TLM scheme may be derved n two dfferent ways. One way s to derve t as a specal and smplfed case of the more general Lattce Boltzmann schemes for the soluton of the Naver-Stokes equatons as was demonstrated by Chopard et al. [4]. Though ths approach opens the way to ncluson of many nonlnear effects of sound propagaton t s not very ntutve. Therefore another dervaton shall be presented here whch reles on a physcal model of the feld effects and was proposed by Kagawa for the acoustc feld [5, 6]. The basc dea s to follow Huygens prncple that states, that each pont on a wavefront can be consdered as a source of secondary wavelets, so that the propagaton can be consdered as a contnuous process of scatterng of elementary component waves. The superposton of all scattered (elementary) waves makes up the wave. Ths mechansm may be dscretzed as follows: The flud s replaced by a network of ppes wth equal dameters whch are jont at fxed dstances x (see fgure 1). The junctons are called nodes. Wthn the ppes acoustc pressure pulses may travel undsturbed at the speed of sound,.e. no dampng or boundary effects are consdered. The dameter of the ppes s assumed to be much smaller than the shortest wavelength of nterest so hgher order ppe modes are neglected. From ths follows, that f a set of acoustc pressure pulses s released at the same tme at arbtrary nodes, these pulses travel through the ppes and reach ther neghborng nodes after a tme nterval of t = x/c 0 wth c 0 beng the speed of sound n the ppe. For dmensons, the juncton may be seen as a meetng of the ppe carryng the ntal, ncomng pressure pulse wth three outgong ppes. Thus the acoustc pressure pulse arrvng at the node sees an apparent enlargement of the cross secton by a factor of three, whch means a correspondng change of the mpedance. At ths mpedance msmatch the ncomng pressure pulse s reflected wth half of the ntal ampltude and a reversal of sgn, whle t s transmtted nto the other three ppes wth half the ntal ampltude and no change of sgn (see fgure 1). From ths pont on the process of travelng n the ppes and scatterng at the nodes s repeated, wth four pulses arrvng smultaneoulsy at each node and each scatterng nto the other three ppes. As the process of scatterng and travelng s lnear, superposton apples and the net result s the sum of the component effects. The algorthm descrbed above can formally wrtten as follows: f ( x + x, t + t) = S j f ( x, t) (1) wth f ( x + x, t + t) representng the ampltude of the pressure pulse that travels from the node at poston x to the node at poston x + x between tmes t and t + t. x s the vector pontng to the nearest neghbor n the -th drecton, and S j s the scatterng matrx S j = Because of the smple structure of the scatterng matrx equaton (1) may also be wrtten as f ( x + x, t + t) = 1 f j ( x, t) f ( x, t) () j on FEM Technology 3

4 % % % & & & % % % & & & % % % & & & x ' ' ' ( ( ( ' ' ' ( ( ( ' ' ' ( ( ( : : : : : : : : : / / / / / / / / / !!! " " "!!! " " "!!! " ; ; ; ; < < < < ; ; ; ; < < < < ; ; ; ; < < < < before scatterng p after scatterng p/ p/ p/ p/ Fgure 1: The acoustc medum s replaced by a network of ppes, n whch pressure waves travel. The junctons are called nodes, the vectors x pont from one node to the nearest neghbors. The scatterng process s depcted n the lower part of the fgure. where s defned such that x = x. The acoustc pressure p ak and the velocty v are calculated from f ( x, t) applyng the followng formulae: p ak ( x, t) = 1 f ( x, t) (3) v( x, t) = 1 ρc x t f ( x, t) (4) Ths scheme s numercally equvalent to a second order Fnte Dfference scheme for the wave equaton, whch s shown as follows: The acoustc pressure at a poston x and tme t + t s gven as Repeated use of eq.(3) yelds p ak ( x, t + t) = 1 f ( x, t + t) p ak ( x, t + t) = 1 p ak ( x + x, t) f ( x + x, t) = 1 p ak ( x + x, t) p ak ( x, t t) + f ( x, t t) Ths may be rearranged to gve p ak ( x, t + t) p ak ( x, t) + p ak ( x, t t) = 1 p ak ( x + x, t) p ak ( x, t) whch equvalent to the correspondng Fnte Dfference equaton f x t = c on FEM Technology 4

5 speed of sound 1500 m/s flud densty 1000 kg/m 3 sze of calculaton area 4 4 mm number of nodes = Spatal resoluton x = 5 µm temporal resoluton t = s membrane complance Pa/m membrane mass kg/m frequency of exctaton 1 MHz Table 1: Parameters of the test calculaton of surface waves. Ths last result means that the speed of wave propagaton c s gven by the rato of the spatal and temporal resoluton. Note that the overall wave speed c s lower by the factor 1/ than the speed of component pressure pulses travelng n the ppes, whch s a feature of the method that s surprsng when frst encountered. 3 The ultrasound transducer array model The ultrasound transducers to be modeled here are capactve transducers made up of a flexble membranes above a fxed counter electrode. If a voltage s appled the membrane s attracted to the counter electrode untl the electrostatc force s balanced by the restorng force of the membrane stffness. Varatons of the magntude of the appled voltage lead to movements of the membrane whch then acts as a sound generator for the flud outsde the transducer. The transducer s modeled as a smple masssprng system where the sprng wth sprng constant D provdes the restorng force and the mass m corresponds to the membrane mass of the real transducer. From the flud sde the acoustc pressure p ak calculated usng eq.(3) acts on the membrane wth area A, also an electrostatc force F el may be appled. The equaton of moton for the membrane s m s = p ak A Ds + F el where s s the deflecton of the membrane. Ths equaton of moton s then solved employng the Newmark tme ntegraton scheme for every membrane and the resultng membrane velocty s appled as a boundary condton to the flud by evaluaton of eq.(4). 4 Results As a frst test a two dmensonal setup s consdered, whch conssts of water as the acoustc medum. The calculaton area s 4 4 mm, three of the surroundng boundares are rgd walls. The membrane array s placed along the bottom boundary. In ths case one membrane model per TLM-node s appled and the membrane mass s chosen to be very small whch corresponds to a free surface wth locally reactng restorng force. Further detals are gven n table 1. The frst 80 membranes on the left are excted wth a snusodal force wth 1 MHz whle all other membranes are allowed to move freely. After turnng on the exctng force two effects clearly are observed: a sound wave s generated by the excted transducer and propagates nto the flud a surface wave s generated that spreads along the boundary of the flud wth lower propagaton velocty than the sound wave on FEM Technology 5

6 Fgure : Pressure feld at tme t = s after turnng on the exctaton force ( MHz). Whte and black colors correspond to hgh and low pressure, respectvely. The surface wave that propagates along the bottom boundary can clearly be seen to be slower (have shorter wavelength) than the sound waves, that spread nto the medum. Repeated experments wth other exctaton frequences show that the typcal dsperson relatons for sound waves c = λf and surface waves c λ are satsfed by ether wave type. Fgure shows the pressure feld at the tme t = s after turnng on the exctaton force, here wth a frequency of MHz. In order to compare the results quanttatvely the same setup s also smulated usng ANSYS. The spatal and temporal resoluton s changed to x = 0µm and t = s to save calculaton tme. Fg. 3 shows the membrane deflectons as a functon of poston calculated wth ANSYS and TLM at tme t = s after turnng on the exctaton force. Although the general shape agrees well for both calculatons the ampltudes show dfferences of up to 0%. The reason for ths dfferences s not yet understood. Although ts too early to compare calculaton tmes t s noteworthy that the TLM-program takes about mnutes on 800x800 grd whle ANSYS needs 0 mnutes to complete on a 100x100 grd. 5 Summary and Conclusons A model for the calculaton of acoustc and surface waves n fluds s establshed on the bass of the Transmsson Lne Matrx (TLM) method. The free surface boundary condton s modeled by masssprng-systems whch are coupled to acoustc feld by the acoustc pressure and velocty. Frst results clearly demonstrate the ablty of ths scheme to capture acoustc and surface wave propagaton effects. Dfferences between the TLM soluton and a reference soluton calculated wth the FE method ndcate, that further checks or refnements may be necessary. The hgh computatonal speed of the TLM-scheme allows 3d problems to be modeled. A 3d verson of the TLM scheme was coded recently but the results have not yet been verfed. Frst tests show that t s possble to handle problems wth up to 4 mllon nodes on a contemporary PC n reasonable tme ( 1 s CPU tme per model tme step). on FEM Technology 6

7 1e 11 FEM TLM 5e 1 0 Deflecton n m 5e 1 1e e 11 e 11.5e Poston n m Fgure 3: Deflecton of the membranes as a functon of poston calculated wth ANSYS (sold lne) and TLM (dashed lne) at t = s after turnng on the exctaton force. References [1] G. Kron, Equvalent crcut of the feld equatons of Maxwell, Proc IRE, vol. 3, pp , [] J. Whnnery and S. Ramo, A new approach to the soluton of hgh-frequency problems, Proc IRE, vol. 3, pp , [3] J. Whnnery and S. Ramo, Network analyser studes of electromagnetc cavty resonators, Proc IRE, vol. 3, pp , [4] B. Chopard and M. Droz, Cellular Automata Modelng of Physcal Systems. Cambrdge Unversty Press, [5] Y. Kagawa and T. Yamabuch, Fnte-element equvalent crcuts for acoustc feld, Journal of the Acoustcal Socety of Amerca, vol. 64, pp , [6] Y. Kagawa, T. Tsuchya, B. Fuj, and K. Fuchoka, Dscrete Huygens model approach to sound wave propagaton, Journal of Sound and Vbraton, vol. 18, no. 3, pp , on FEM Technology 7