Audio Engineering Society. Convention Paper. Presented at the 133rd Convention 2012 October San Francisco, CA, USA

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1 Audo Engneerng Socety Conventon Paer Preented at the 33rd Conventon October 6 9 San Francco, CA, USA Th Conventon aer wa elected baed on a ubmtted abtract and 75-word rec that have been eer revewed by at leat two qualfed anonymou revewer. The comlete manucrt wa not eer revewed. Th conventon aer ha been reroduced from the author' advance manucrt wthout edtng, correcton, or conderaton by the Revew Board. The AES take no reonblty for the content. Addtonal aer may be obtaned by endng requet and remttance to Audo Engneerng Socety, 6 Eat 4nd Street, New York, New York 65-5, USA; alo ee All rght reerved. Reroducton of th aer, or any orton thereof, not ermtted wthout drect ermon from the Journal of the Audo Engneerng Socety. Study of the nteracton between radatng ytem n a coaxal loudeaker Eí, A., Cárdena, W.A., Martínez, J., Ram, J., Carbajo, J. Acútca Beyma, S.L., Moncada, Valenca, 463, San alejandro.e@beyma.com, j.martnez@beyma.com IUFACT, Unverdad de Alcante, 38, San wac@alu.ua.e, jram@ua.e, jeu.carbajo@ua.e ABSTRACT In th work, the rocedure followed to tudy the nteracton between the md and hgh frequency radatng ytem of a coaxal loudeaker exlaned. For th uroe, a numercal Fnte Element model wa mlemented. In order to ft the model, an exermental rototye wa bult and a et of exermental meaurement, electrcal medance and reure frequency reone n an anechoc lane wave tube among thee, were carred out. So a to take nto account the dlacement deendent nonlnearte, a dfferent nut voltage arametrc analy wa erformed and nternal acoutc medance wa comuted numercally n the frequency doman for ecfc hae lug geometre. Through nverely tranformng to a tme dfferental equaton cheme, a lumed element equvalent crcut to evaluate the mutual acoutc load effect reent n th tye of acoutc couled ytem wa obtaned. Addtonally, the croover frequency range wa analyzed ung the Near Feld Acoutc Holograhy technque.. INTRODUCTION In th work, the rocedure followed to tudy the nteracton between the md and hgh frequency radatng ytem of a coaxal loudeaker exlaned. The mutual acoutc load wa tuded baed on a lumed arameter nonlnear model that take nto account the effect that the adabatc roce n the comreon chamber caue over the dahragm ([] and []), a well a the mechancal and electrcal nonlnearte on the tffne of the uenon, the force factor and the voce col nductance. For th uroe, a numercal Fnte Element model wa mlemented. In order to ft the model, an exermental rototye wa bult, contng of a 6 nche annular mdrange

2 loudeaker and a nche tweeter attached to a folded duct. Th ytem acoutcally loaded wth a hae lug n the cae of the mdrange eaker and a bullet ece whch n turn form the nner art of the folded duct. The ntal numercal model arameter were ftted from electrcal medance and reure frequency reone meaurement n the mall gnal regme. So a to determne the ndeendent comonent (comreon chamber, roagaton ath, geometry) effect on the whole ytem oeraton, each loudeaker wa meaured earately n an anechoc lane wave tube. Alo, and o a to dentfy the contrbuton of the man vbraton mode of the dahragm, a numercal modal analy wa carred out, ung the Young modulu and lo factor data of the membrane and uenon materal rovded exermentally by alyng drect and nvere technque. Once the lnear model wa roerly ftted, a dfferent nut voltage arametrc analy wa erformed, makng t oble to etablh the large gnal regme behavor of both tranducer (mdrange and tweeter). Snce a lumed arameter model doe not contemlate ecfc geometre, the acoutc radaton medance reented by each hae lug wa calculated numercally and nverely tranformed to the tme doman. Baed on the numercal oluton of tme doman dfferental equaton, the dynamcal behavor of the dahragm and the nonlnear ar comreon n the comreon chamber were obtaned. The ueroton of the reone of each loudeaker wa comared wth that meaured exermentally to etablh the effect of the mutual coulng acoutc load and to evaluate the dtorton generated for varou nut voltage level. Furthermore, the croover frequency range of the whole radatng ytem wa analyzed ung the Near Feld Acoutc Holograhy (NAH) technque, wdely ued n the characterzaton and aement of loudeaker erformance. The tructure of the aer a follow; n ecton the bac concet that uort the lumed arameter and decrbe the behavor of the eaker model are brefly exlaned, wth ecal emha on nonlnear contrbuton. In ecton 3, the aroach ued to addre each of the roblem that are are exlaned, ecfcally, the reult of exermental meaurement are exlaned and dcued. We reent the FEM numercal model mlemented to calculate the acoutc medance load on each of the eaker, whch are determned by the ecfc geometry of the hae lug and horn. The block dagram of the alcaton mlemented to olve the et of dfferental equaton n tme doman alo reented. In Secton 4, the reult that valdate the model and the ytem reone are tuded, a well a the effect of dfferent change n both geometre, wth ecal emha on the tranton zone between both radatng ytem. The effect of an ncreae of the nut voltage n the non-lnear reone (THD %) alo nvetgated. In order to tudy the radaton of coaxal eaker n the croover regon, ome reult of the Near-Feld Acoutc Holograhy (NAH) meaurement are reented. Fnally, ecton 5 decrbe the man concluon of th work.. CONCEPTS Dfferent method baed on the rncle of electrodynamc loudeaker have been develoed to model horn drver and comreon drver, added the effect due to ar layer formed n the comreon chamber and hae lug, a well a the nonlnear roagaton nde the hae lug and the comlex medance of the horn ([], [], [3] and [4]). The method ued to develo the reent tudy baed on the numercal oluton of a et of dfferental equaton that decrbe the reone of the loudeaker to any tmulu lke noe, ne-wee or muc. A well known, the loudeaker a nonlnear ytem, where t major nonlnearte deend on the voce col dlacement and current ([5] and [6]). The electro mechanc mechanm of a loudeaker can be dvded n two art: electrcal and mechancal, both of them couled by the force factor Bl( x ). The dfferental equaton whch decrbe the electrcal art are: dl ( e( xt )( )) dl ( ( x ) ( t)) dx () ut () = tr () e Blx ( ) dt dt dt dl ( ( x ) ( t)) = (() t ()) t R( x) () dt where ut () the nut voltage and t () the current n the voce col, both of them are tme deendent, R e the dc retance of the voce col, Le ( x) the voce col nductance and deend on the voce col dlacement x ; L ( x ), () R x decrbe the eddy current t and ( ) AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page of

3 at hgh frequence, baed on the LR- model dcued n [7], and ( ) Bl x the force factor. Regardng the mechancal art, a nonlnear term n the comlance aocated to the comreon chamber couled to the hae lug and the rear chamber mut be revouly exlaned. A t known, one of the comonent that add dtorton to the acoutc outut the nonlnear relatonh of the reure and volume change n thee two chamber. Th relatonh can be exreed by the adabatc equaton a: γ PV = ( P + )( V V) (3) γ where the acoutc reure n the chamber, V the chamber volume, P the atmoherc reure, γ adabatc contant (n ar, γ.4) and V rereent the ntantaneou varaton of the chamber volume, that deend on the dahragm dlacement, beng V = Sd x, where S d and x are the urface and dahragm dlacement reectvely. Gven that the comreon chamber of a drver not a comletely cloed encloure and that connected wth the outde regon through the hole n the hae lug and throat of the horn, n [] hown that: γ Sdx Sx xx (, ) = P V To tudy the behavor of the comreon drver, equaton (4) hould be combned wth the dfferental equaton ytem that balance force and voltage that govern the dahragm movement. The Taylor exanon of (4) can be tranformed nto dfferental equaton f the nut medance of the hae lug and horn aumed to be the acoutc medance ρ c of a lane wave that roagate n ar, beng ρ the denty and c velocty of ound n ar reectvely. Gven that th hyothe not alcable n our cae, neceary to ue a et of equaton caable of takng nto account the comlex acoutc load of the hae lug and the ecfc horn. (4) Accordng to Vohvllo n [], the acoutc medance load of the hae lug and horn can be aroxmated by a fractonal-ratonal functon a: n () + b + b ρc () = + + Z () = = v a a beng () the chamber ound reure and v () the velocty of the ar n the hae lug, both exreed n the Lalace doman, and b, b, a and a the functon coeffcent. By erformng an nvere tranform oble to convert the acoutc medance n a dfferental oerator and to obtan a et of dfferental equaton that take nto account the comlex medance of the hae lug and horn. (5) The dlacement n the hae lug x can be exreed n term of the reure n the chamber a: n n n dx d d d = n + g n n n + g+ g dt ρc dt dt dt n n d x d x dx h n h n... h n n dt dt dt where g and h are the adjutment coeffcent. If x and x are ubttuted n equaton (4) and after the reure n the comreon chamber and t dervatve are relaced n equaton (6), th exreon can be n-tme ntegrated numercally to obtan x. (6) In th work, the acoutc medance aroxmated by the tranfer functon: + b + b + + Z () = ρc a a (7) where b, b, a and a are the adjutment coeffcent needed to aroxmate th tranfer functon to that obtaned numercally ung fnte element. Then equaton (6) can be rewrtten a: d x d x dx 3 d d = 3 + a + a b b dt ρc dt dt dt dt (8) In th cae, the dervatve of the reure n the comreon chamber are obtaned analytcally. Regardng the mechancal decrton of the loudeaker dlacement, the force equlbrum AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 3 of

4 equaton [3] reented addng an addtonal term S d that decrbe the comreon chamber reone to the dahragm movement: Bullet ece [ ] d x m Bl( x) ( t) Fm( x,, ) = m + dt dx dx + Rm + Rmv ( x) + R + dt dt + K ( x) + K ( x) x( t) + S m r d (9) where R m the ytem mechancal retance, Rmv ( x ) refer to the thermo vcou loe reent n the comreon drver and R tand for the loe caued by the turbulence n the hae lug, whle Km ( ) ( ) x and Kr x correond to the uenon and rear chamber tffne reectvely. The reluctance force F (,, ) m x can be modeled n an aroxmated manner a: Fgure : Prototye bult: (a) cture and (b) croectonal vew. For th uroe, ound reure meaurement were reformed n a lane wave tube wth anechoc termnaton a that hown n Fgure. () t dle ( x) () t dl( x) Fm (,, x ) = dx dx () Th nonlnear dfferental equaton ytem olved ung a Runge-Kutta algorthm n a block orented mlementaton a wll be hown n ecton APPROACH Th ecton decrbe the ecfc aroach ued to addre each of the key ont of the roce. The frt te wa to erform ome meaurement on a rototye bult for th uroe. Then, the bac data uled to the FEM model mlemented to obtan the acoutc medance, and fnally the block dagram of the alcaton develoed to olve the dfferental equaton reented. 3.. Exermental meaurement Fgure how a cture and a cro-ectonal vew of the rototye bult ued to calbrate the model. Fgure : Plane wave tube wth drver. To dentfy and characterze the frequency range over whch the Tweeter (TWT) ha an mortant nfluence, ome meaurement of the frequency reone were carred out. In a frt te, the olarty of the exctng gnal of the tweeter wa nverted. Fgure 3 how the effect of th olarty nveron n the coaxal loudeaker. A t can be een, the cancellaton effect more evdent for frequence located n a very wde range (between 3 khz and 7 khz), where varaton reach level of over 7 db, a n the cae of 6 khz. The area between 7 khz and 9 khz reent a d of more than db wth reect to the adjacent band for both cae even though not n the tranton zone of the eaker. AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 4 of

5 SPL (db ref. upa) No Inverted Inverted Fgure 3: Effect of the hae nveron n the coaxal eaker SPL. In a econd te and o a to analyze the nteracton between common cavte nvolved n the acoutc ath of the ytem, the folded duct ecton correondng to the mdrange eaker wa covered wth latc ma reducng the acoutc load volume of the tweeter dahragm. Fgure 4 how a comaron between the tweeter frequency reone for the normal confguraton and the modfed one. attenuated, a n the cae of 6 khz. Thee and other effect can be tuded wth the model mlemented n th work, nce t can take nto account change n the geometry and the acoutc medance load of the eaker, beng of great nteret n the degn roce of thee tye of eaker. 3.. Comutaton of the acoutc medance wth Fnte Element Analy One way to ntegrate nto the analy of the eaker erformance ome factor lke the comlex geometre of the hae lug and duct on whch the eaker radate, to numercally calculate the acoutc medance load of each dahragm. Th numercal exerment erformed n harmonc regme ung the Fnte Element Method. The axymmetrc model develoed cont of 48 trangular element for a dcretzed cro ecton of 696 mm comrng comreon chamber, rear chamber and a folded duct condered a mutual load for both dahragm. Fgure 5 how the reult of the numercal model at 3 khz, where the radaton boundary condton and PML regon are roerly dected. SPL (db ref. upa) 3 9 Radaton boundary condton PML regon Covered Uncovered Fgure 4: Effect of the acoutc ath modfcaton n the tweeter SPL. By mean of an analy of Fgure 4, an ncreae of the SPL due to the renforcement of the acoutc energy roagaton through an only ath evdenced, but t can be alo arecated that ome of the reonance frequence aocated to the covered cavty are Fgure 5: Detal of the axymmetrc numercal model harmonc analy reult at 3 khz. Once the acoutc medance calculated, the tranfer functon Z() ued n the dfferental equaton model adjuted. Fgure 6 how the acoutc medance n the cae of the Mdrange eaker and the Tweeter n a frt aroxmaton, where the acoutc medance comuted a the rato of the averaged reure and artcle velocte n the roxmte of the dahragm. It AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 5 of

6 NO LINEAL ACUSTICA PARTE MECANICA PARTE ELECTRICA Scoe L (Lec/(+*ex(Lec3*(u-x)))+Lec) /(rho*c) Gan a a (Lec/(+*ex(Lec3*(u-x)))+Lec) Dvde d3xt/dt3 dxt/dt dxt/dt xt d/dt d/dt (Pa) /(Mm) dx/dt dx/dt ((Lec3*Lec*ex(Lec3*(u-x)))/((+ex(Lec3*(u-x)))^)) d/dt Dvde b b P x x (m) AQUI VENDRIA SUMAR LOMISMO PERO L I ((Lec3*Lec*ex(Lec3*(u-x)))/((+ex(Lec3*(u-x)))^)) P*(V^Y)*Y P*(V^Y)*Y (Rex+R)*u AQUI VENDRIA I* R ARRIBA SUMAR RM+R dx/dt dx/dt dx/dt ((Lec3*Lec*ex(Lec3*(u-x)))/((+ex(Lec3*(u-x)))^)) R*u / R V^Y Y+ (/(Kmc*ex((-(u-x)^)/(*Kmc^)))) (Sef^)*P*(Vr^Y)*Y R*u (Bl*ex(-Blc*(u+Blc)^)) u^ Rt*u Rm x*u Rm v*u Sef Fcn u^(-y-) (Bl*ex(-Blc*(u+Blc)^)) / Fcn u^(-y) (Vr - Sef*u)^(-Y-) u^ u^(-y-) u^ u^(y-) dldx ((Lec3*Lec*ex(Lec3*(u-x)))/((+ex(Lec3*(u-x)))^)) Q Q Vd (A) Scoe (A) Sef St V Sef St Sef St x dxdt dx/dt U n (V) x (m) U n (V) From Workace ulo (Pa) (Pa) (A) x (m) U n (V) u To Workace To Workace x To Workace3 To Workace4 xt To Workace5 v To Workace6 To Workace7 Scoe Eí et al. Study of the nteracton... mut be emathzed that the acoutc medance value reent mortant devaton deendng on the oton condered for the calculaton Block-Orented ytem for olvng the Dfferental Equaton The block dagram ued to olve the nonlnear dfferental equaton ytem n the tme doman hown n Fgure 7, where acoutc, mechancal and electrcal art a well a the varable that relate them, are hown. 6 Zt+Z (db) FEM Average Aroxmaton Zt+Z (db) FEM Average Aroxmaton Fgure 6: Acoutc medance for the Mdrange eaker (u) and the Tweeter (down) comuted and ftted wth a, a, b and b coeffcent. It hould be noted that varaton n the medance for the numercal calculaton nvolve more than a ngle reonance along the frequency range, whch cannot be adjuted wth great accuracy by thee coeffcent. To acheve a behavor of th tye neceary to handle a tranfer functon of hgher order, mong dervatve of thrd or fourth order n the dfferental equaton ytem develoment. Fgure 7: Block-dagram ued to olve the nonlnear dfferental equaton ytem (gray: acoutcal, darkgray: mechancal and black: electrcal) 3.4. Swet-Sne baed lnear and nonlnear analy In order to tudy the dtorton behavor of the comreon drver, a wdely ued non-lnear ytem characterzaton methodology baed n Exonental Swee Method wa emloyed. Th accomlhed by generatng a gnal that vare n frequency over tme. A exected, wth th tye of tmulu t can be only AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 6 of

7 tuded the harmonc dtorton. The nut gnal defned a: reoluton of 4 mm n a arallel lane laced 5 mm n front of the comreon drver outut. c T t xt () n T = ω e c () Fgure 8 how the exermental etu ued for the NAH meaurement. where t [, T] duraton and c ln ( ω / ω ) the tme nterval, T the gnal =, wth ω and ω the ne wet tart frequency and fnal frequency reectvely. A weakly nonlnear ytem excted by th gnal roduce a reone gnal yt () wth hgher order harmonc. In order to comlete the ytem dentfcaton, the outut gnal yt () convolved wth an nvere wee x () t whch atfe x() t x () t = δ () t, obtanng: () t = y() t x () t () The nvere wee comenate for the grou delay and the tmulu magntude ectrum, and can be obtaned from the nut gnal alyng the Fourer tranform [8]: X ( ω) X ( ω) = X ( ω) (3) where X ( ω ) the comlex ectrum of the nut gnal x() t and X ( ω) the ectrum of the nvere wee x () t. Fgure 8: Exermental etu ued for the NAH meaurement. 4. RESULTS 4.. Valdaton of the model Th ecton exlan n detal the roce undertaken to determne the man feature of the model. Fgure 9 how the curve of meaured and mulated reure of both eaker for an nut gnal of 7 mv Radaton tudy by mean of NAH The Near-Feld Acoutc Holograhy (NAH) technque a rocedure ued to recontruct the ound feld and the vbraton velocty of an object or ound ource from meaurement wth mcrohone n a lane arallel and cloe to the ource (hologram lane). Th technque had t orgn n a work by Wllam and Maynard n 985 ([9] and []), that mroved conventonal holograhy by near-feld meaurement, whch allow to cover a wde frequency range and alo to cature the evanecent wave (ubonc wave that decay exonentally wth dtance from the ource []) created by the ound ource and that contan hgh reoluton nformaton about the ource ([]). In the reent work, the meaurement grd ued had a atal SPL (db ref. upa) Model Mdrange Model Tweeter Meaured Mdrange Meaured Tweeter 3 4 Fgure 9: Meaured and mulated SPL frequency reone of the coaxal loudeaker. AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 7 of

8 Note that the model take nto account the mot rereentatve charactertc of the real drver unt. For th reaon, ome of the aect related to the membrane tructural behavor cannot be roerly modeled. The global ytem reone reented n Fgure. It can be arecated a lght good agreement between comuted and meaured data n the acoutc croover regon. SPL (db ref. upa) 8 6 chamber to avod the roll-off n the hgh frequency range to be o abrut or to modfy the tweeter o that the reonance frequency decreae to the a band mmedately next to the Mdrange decay. Although both oblte are feable, the degn of a ytem of th tye nvolve a comrome between effcency and "flatne-moothne" of the acoutc reone. To tudy th henomenon wthout the need to hycally modfy the rototye, dfferent mulaton of the model mlemented are erformed for dfferent value of ma m m and volume of the comreon chamber V, whch add rgdty to the ytem changng the reonance frequency of the tweeter. Fgure and how the reult of the mulaton for dfferent value of ma and volume, where dcontnuou lne rereent the Tweeter and Mdrange loudeaker ndvdual reone reectvely. 4 Model Exermental Fgure : Comaron between comuted and meaured SPL. SPL (db ref. upa) Aement of the ytem behavor n the tranton band A frt aroach to the roblem would conclude that the tranton zone of the eaker controlled by mechancal arameter uch a the movng ma of the tweeter, the comlance of the uenon and ome other geometrcal factor a the ar volume cavte and urface of the duct attached to the vbraton of the half-nch eaker. The hgh frequency regon more affected by the geometry that form the bullet ece wth the folded duct. The effect of hae nveron of the ytem, dcued n ecton 3. (Fgure 3), allow to determne the ectral regon that are more affected by the detructve nterference wthn th folded duct. Th nformaton ueful a an ntal crteron of otmzaton of the radatng ytem and an alternatve that can mrove the frequency reone n regon above the tranton zone, controlled by the hgh frequency tranducer. In a frt aroxmaton, the oluton to th roblem could be to ncreae the volume of the comreon m m.8*m m Fgure : Effect n the overall reone of the modfcaton of the m of the tweeter. SPL (db ref. upa) m.*v Fgure : Effect n the tranton zone of the varaton of the V of the comreon chamber. V AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 8 of

9 An ncreae of the movng ma of the tweeter mle a decreae of the reonance frequency that make the a band to be otoned nearer to the decay of the Mdrange and thereby to acheve a hgher gan when t almot dulcated. In the other hand, an ncreae of the volume of the comreon chamber make the reone curve to be hfted to hgher frequence THD (%) In th ecton, an analy of harmonc dtorton generated by the eaker carred out from an exermental and mulaton aroach. Snce the model baed on the numercal oluton of a et of dfferental equaton n the tme doman, the exctaton gnal can be any tme equence amled at the ame rate a the tme te of the algorthm. In th cae the nut gnal generated wth a amlng rate of 88 khz, due to the temoral reoluton requred to fx the model. Once the gnal for the tate varable uch a reure, velocty and dlacement current are obtaned, a decmaton roce erformed o a to work at the ame amlng rate that n the exermental meaurement. Amltude (Pa) Amltude (Pa) Tme () Tme () Fgure 5: Model lnear reone and hgher order harmonc for two dfferent nut voltage: 8 mv (u) and 9 mv (down). The wet ne ued ha a length of.5 and cover a frequency range from Hz to khz. Fgure 5 how the outut gnal convolved wth the nvere wee for nut voltage of 8 mv and 9 mv. It mut be remarked that an ncrement of th nut voltage nvolve the hgher order harmonc aearance due to the nonlnear dtorton moed by the electrcal and mechancal arameter a well a the adabatc comreon n the comreon and rear chamber. In Fgure 6, frequency reone (FR) and harmonc dtorton are hown for two nut voltage. It can be arecated that the meaured harmonc dtorton more ronounced for the 3 th order harmonc (H3) than for the nd order harmonc (H) when the nut voltage ncreaed from 8 mv to 9 mv. SPL (db ref. upa) mv H H3 9 mv H H3 3 4 Fgure 6: Meaured harmonc dtorton ncrement for two nut voltage: 8 mv and 9 mv NAH reult A t wa dcued n ecton 3., the geometrcal hae of the bullet ece and the folded duct can be otmzed o a to mrove the ytem erformance n the hgh frequency range. The NAH technque allow to tudy n detal the effect on the ytem reone of changng thee geometre. Fgure 7 how the ound reure n a lane of x mm cloe to the drver outut for dfferent frequence nearby the tranton range. It mortant to remark that for 4.5 khz (Fgure 7.a), the radaton manly domnated by the mdrange loudeaker; for 5.8 khz (Fgure 7.b) the ytem n the tranton range and for 6. khz (Fgure 7.c) the tweeter the rncal acoutc ource. Fnally, for 7.4 khz (Fgure 7.d) the duct mode are evdenced. It can be arecated that the ytem radaton for frequence bellow 6 khz can be aroxmated a a AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page 9 of

10 lane wave raagated n a duct, nce the radu of the mouth of the comreor drver (5. mm) lower than the wavelenght roagatng nde of t. The model ha been contrated wth exermental data and can be a owerful tool to reduce tme and cot n the degn roce and to redct qute cloely the frequency reone and harmonc dtorton of the radatng ytem condered. It alo ncororate a module that allow hearng the effect of the varaton of any arameter n order to check, n a frt aroxmaton, f mall change n thee hycal arameter are aurally ercetble. (a) (b) Bede the exermental and numercal tudy, an analy of the nteracton between the eaker wa erformed by alyng the NAH technque. The reult reveal nformaton about the wave roagaton nde the duct, mortant for the nternal geometry otmzaton and to mrove the acoutc medance coulng. (c) (d) 6. REFERENCES [] A. Vohvllo, Nonlnearty n Horn Drver-Where the Dtorton Come From?, 3 th AES conventon, Lo Angele, Oct 5-8, [] W. Klel, Nonlnear Sytem Model for Horn Loudeaker, 99 th AES conventon, New York, Oct 6-9, 995 (e) Fgure 7: Sound reure atal dtrbuton at the hologram lane for dfferent frequence: (a) 4.5 khz, (b) 5.8 khz, (c) 6. khz and (d) 7.4 khz. (e) Prototye frontal vew. For frequence above 7 khz, the roagaton n the folded duct doe not accomlh th crtera anymore, and hence the nternal geometry mut be modfed to reduce the nfluence of modal behavour and to mrove the outut load medance matchng. 5. CONCLUSIONS In th aer wa decrbed the roce carred out to adat a numercal model etablhed n the lterature to the tudy of a coaxal eaker. Th model ncororate ecfc nformaton about the geometry of the radatng ytem by numercally modelng the load medance t reent to each loudeaker. [3] H.Schurer, A. Berkhoff, Modelng and Comenaton of Nonlnear Dtorton n Horn Loudeaker, 96 th AES conventon, Amterdam, Oct 6-9, 995 [4] A.N. Thele. Loudeaker n vented boxe, art and. Journal of Audo Engneerng Socety,. 9:38-39, , 97. [5] W. Klel, Aeng Large Sgnal Performance of Tranducer, Klel GmbH [6] W. Klel, Loudeaker nonlnearte - caue, arameter, ymtom, J. Audo Eng. Soc., vol. 54, , 6. [7] M. Dodd, W. Klel, and J. Oclee-Brown. Voce col medance a a functon of frequency and dlacement. AES Conventon:7 th. Audo Engneerng Socety, October 4. [8] M. Potr, P. Balaz, and B Laback, Multle Exonental Swee Method for Fat Meaurement AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page of

11 of Head-Related Tranfer Functon. J. Audo Eng. Soc., vol. 55, No. 7/8, 7 July/Augut. [9] J.D. Maynard, E.G. Wllam, y Y. Lee, Nearfeld acoutc holograhy: I. Theory of generalzed holograhy and the develoment of NAH, Journal of the Acoutcal Socety of Amerca, Vol.78, N_ 4, , (985). [] E.G. Wllam y H.D. Dardy, Nearfeld acoutcal holograhy ung an underwater, automated canner, Journal of the Acoutcal Socety of Amerca, Vol. 78, N_, , Augut (985). [] E.G. Wllam, Fourer Acoutc Sound Radaton and Nearfeld Acoutcal Holograhy, Ed.Academc Pre, (999). [] Wllam, Numercal evaluaton of the radaton from unbaffled, fnte late ung FFT, Journal of the Acoutcal Socety of Amerca Vol. 74, July (983). AES 33rd Conventon, San Francco, CA, USA, October 6 9 Page of