10ème Congrès Français d'acoustique Lyon, Avril 2010

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1 1ème Congrès Français d'acousique yon, 1-16 Avril 1 Modeling of he nonlinear sorion in elecrodynamic loudspeakers caused by he voice-coil inducance Andrzej Dobrucki 1, Benoi Meri, Valérie emarquand, Guy emarquand 1 Wroclaw Universiy of Technology, Wybrzeze Wyspianskiego 7, 537 Wroclaw, Poland, andrzej.dobrucki@pwr.wroc.pl aboraoire d'acousique de l'universié du Maine, UMR CNRS Avenue Olivier Messiaen e Mans Cede 9, France, {benoi.meri.eu, valerie.lemarquand, guy.lemarquand}@univ-lemans.fr The elecrodynamic loudspeaker is a srongly nonlinear sysem. The main causes of nonlineariy are he nonlinear siffness of he suspensions and he nonuniform sribuion of he magneic flu densiy along he air gap in he magneic circui. A hird cause, he nonlinear voice-coil inducance is ofen underesimaed. The elecrical impedance of he voice-coil in he low frequency range is considered usually as a pure resisance, and he influence of is inducance is negleced : his is a reason of he underesimaion of is nonlineariy. The voicecoil inducance depends on boh he splacemen of he voice-coil and he curren inensiy flowing hrough i. Adionally, he relucance force proporional o he square of he curren appears in he mechanical par of he sysem and depends on he voice-coil splacemen oo. This paper sues he influence of he voice-coil nonlineariy. The fferenial equaion sysem has been derived and is solved using numerical mehods. The harmonic sorions as well as he inermodulaion ones have been compued. The resuls show ha he influence of he voice-coil nonlineariy is significan paricularly for inermodulaion sorions. This influence is weaker han he influence of he force facor nonlineariy, bu sronger han he influence of he suspensions nonlineariy. The influence of he fferen erms in he nonlinear fferenial equaion sysem is also esed. 1 Inroducion The nonlineariies of elecrodynamic loudspeakers lead o harmonic and inermodulaion sorions in he sound producion. Their sources have been sued and described by many auhors, [1] - [1]. Among hem, he suspensions and he moor are prominen sources. The suspensions creae now well-known nonlineariies, and he manufacurers have developed maerials and designs o minimize heir drawbacks. Some alernaive designs have also been proposed which suppress he classical suspensions [11] - [1]. Oher imporan cause of sorion is he nonlineariy of force facor Bl. I srongly depends on he splacemen of moving sysem. The nonlineariy of a voice-coil inducance is usually underesimaed as he source of nonlinear sorion. In low frequency range he inducance is usually negleced in he equivalen circui of he loudspeaker as well as in he analysis. However, if in he eciing signal he low and high frequency componen appear, he nonlineariy of voice-coil inducance produces very annoying inermodulaion sorion. The aim of his paper is o analyse he sorions produced by he voicecoil inducance and compare hese sorions wih sorion produced by he suspension and Bl-produc nonlineariy. Modeling of elecrodynamic loudspeaker wih inducance and eddy currens The elecrical equivalen circui of an elecrodynamic loudpeaker is presened in Fig. 1. I consiss of hree pars: elecrical which is represened by he voice coil resisance R and inducance ; ransducer represened by a gyraor wih gyraion consan Bl (called also force facor) and mechanical represened by mass of he moving sysem, siffness k and mechanical resisance of he suspension r. The voice coil inducance is no a pure inducance. The eddy currens induced in he magneic circui of he loudspeaker reac wih he curren in he voice-coil and cause he decrease of he inducance and he appearance of a real par of he impedance. The influence of eddy currens can be represened by he resisor R μ conneced in parallel wih voice-coil inducance. However, i is a coarse simplificaion. The equivalen circui of he loudspeaker is presened in Fig. 1. U() R Bl r k R μ Fig. 1. lecrical equivalen circui of he loudspeaker wih he inducance of he voice-coil and eddy-curren resisance The calculaion has been conduced for following consan parameers: R 3.3 ohm,.9 kg, r 1.1 Ns/m, k 769 N/m, Bl 5.5 Tm, R μ. ohm. inear erm of he inducance was.17 H. The eciaion was sinusoidal, and he volage ampliude was 1 V. The sound pressure has been compued on he ais a he sance of d1m from aphragm cener, he ameer of he aphragm was equal o. m. The loudspeaker was

2 described wih a sysem of ornary fferenial equaions. The calculaion was realized using Mahcad14 1 workshee. Figures and 3 presen SP and loudspeaker impedance for linear parameers. Frequency [Hz] Fig.. Frequency response of he loudspeaker wih inducance and eddy curren resisor bu wihou nonlineariy Z [ohm] Frequency [Hz] Fig. 3. lecric impedance modulus vs. frequency of he loudspeaker wih inducance and eddy curren resisor bu wihou nonlineariy For higher frequencies he resisor R μ blocks he inducance and he impedance curve becomes fla in his frequency range. However, he oal elecrical resisance is he sum of R and R μ, and he sensiiviy decreases. The resuls of modeling show ha he model of eddy currens based on one resisor blocking he inducance is oo simple. 3. Model of loudspeaker wih inducance depenng nonlinearly on he elecric curren The voice coil inducance nonlinearly depends on elecric curren and on he splacemen. In his secion he model of he loudspeaker wih he inducance depenng on he curren flowing in he voice-coil is considered. The nonlinear dependence of he inducance on he elecric curren is modeled wih polynomial of he second order: () i ( ai ) 1 bi (1) In he elecrical side of he sysem he derivaive of magneic flu appears: Then, he effecive nonlineariy is higher han he nonlineariy of he inducance only. The fferenial equaion sysem for a loudspeaker wih nonlinear inducance depenng on he curren only has he form: The soluion of his equaion sysem are funcions of ime: curren, splacemen and velociy. On he basis of he hird equaion of (3) he acousic pressure can be compued. The acousic pressure a he sance r (r was assumed 1 m) is given by he formula [13]: ωvs S dv p ρ ρ (4) πr πr where: ρ air densiy, S surface of he aphragm. Afer solving equaion sysem (3), he righ hand side of hird equaion of his sysem is pu ino equaion 4 as he acceleraion. In Figs. 4 and 5 he eamples of sound pressure and curren for frequency Hz and volage ampliudes U 1V and 1 V are presened. p () p () dψ d [ () i i] ( 1 ai bi ) 3 d Fig. 4. Acousic pressure vs. ime for frequency f Hz. Upper: U 1V, ower: U 1V i ( 1 ai bi ) U () R i Blv 3 d v dv r v k Bl i () (3) 1 Mahcad14 is he Trade Mark of Parameric Technology Corporaion

3 i1() i1() Fig. 5. Curren in he voice-coil vs. ime for frequency f Hz. Upper: U 1V, ower: U 1V The calculaions have been carried ou for following nonlinear coefficiens: a-1.68, b7.58. The soluions depend srongly on he ampliude of eciing volage. The specra of sound pressure levels for boh eciaion ampliudes are presened in Figs. 6 and 7. clearly visible. Of course, for high eciaion level he sorion producs are also high. 4. Model of loudspeaker wih inducance depenng nonlinearly on he splacemen In his secion he dependence of he voice-coil inducance is modeled by he linear funcion: ( ) ( 1 A) A he elecrical side of he loudspeaker sysem he derivaive of he magneic flu has he form: dψ d [ ( ) i] d d i d where derivaive of splacemen is he velociy. Then, he equaion sysem has he form: ( 1 A) U () Ri Blv A i v d v (5) (6) (7) dv r k Bl 1 A v i i Fig. 6. Harmonic sorion of he acousic pressure for he sysem wih nonlinear inducance (U 1V) The las erm in hird equaion of sysem (7), depenng on he square of a curren, is called a relucance force [14]. In fac, he inducance is nonlinear wih regard o he curren and he splacemen. One of he consequence of he dependence wih he splacemen is he fac ha an adional erm appears in he force applied o he moving par: he relucance force is added o he aplace force. The value A for calculaions is equal o -5 m -1. Fig. 8 presens harmonic sorions creaed by nonlinear inducance for eciaion frequency Hz and volage 1 V, compued wih Mahcad14 workshee. Fig. 7. Harmonic sorion of he acousic pressure for he sysem wih nonlinear inducance (U 1V) The oupu signal for eciaion volage 1 V seems o be unsored, however in Fig 6 he sorion componens are Fig. 8. Harmonic sorion of he acousic pressure for he sysem wih inducance nonlinearly dependen on splacemen

4 I is clearly visible ha he sorion level is very low. Only he second harmonic appears and is level is 5 db lower han he level of fundamenal componen. However, he nonlinear inducance produces for an acual signal no only harmonic componens bu also he inermodulaion componens. In he simples case, when he sysem is ecied wih componens of low and of high frequency, he inermodulaion produc appears as he componen of sum and fference componens. In Fig 9 he inermodulaion sorion produced by he sysem described wih equaion sysem (7) is presened. The sysem was ecied wih wo componens: frequency 5 Hz and volage 1 V, and frequency 15 Hz and volage.5 V. The resul of calculaion is presened in Fig. 9. Fig. 11. Conribuion of he nonlinear erm a he righ hand side in he firs equaion of he sysem (7) in he inermodulaion sorion produced by he sysem wih nonlinear inducance. ciaion he same as in Fig. 9 Fig. 9. Inermodulaion sorion produced by he sysem wih nonlinear inducance. ciaion: f 1 5 Hz, U 1 1 V, f 15 Hz, U,5 V. The inermodulaion sorions are very annoying in he human percepion. I is ineresing o es he conribuion of all nonlinear erms in he equaion sysem (7). The resuls of his esing is presened in Figs Fig. 1. Conribuion of he nonlinear erm a he lef hand side in he firs equaion of he sysem (7) in he inermodulaion sorion produced by he sysem wih nonlinear inducance. ciaion he same as in Fig. 9 Fig. 1. Conribuion of he nonlinear erm a he righ hand side in he hird equaion of he sysem (7) (he relucance force) in he inermodulaion sorion produced by he sysem wih nonlinear inducance. ciaion he same as in Fig. 9 The mos imporan cause is he main nonlineariy of he inducance. I is ineresing o mark ha he level of inermodulaion produc of he second order f -f 1 and f f 1 is even higher han his level when all nonlinear erms are aken ino accoun (compare Figs. 9 and 1. The relucance force produces relaively high level of he second harmonic of he higher componen of eciaion. The mied nonlinear erm curren velociy produces he lowes sorions levels. 5. Comparison of inermodulaion sorion produced by nonlineariy of Bl, siffness of suspensions and inducance, depenng on splacemen In his par ranking of nonlineariies of hree physical parameers depenng on he splacemen has been esed.

5 The esed parameers are: Bl-facor, siffness of he suspensions and voice-coil inducance. The res values of hese parameers were chosen o be he same as in previous pars, i.e..17 H, Bl 5.5 T m, k 769 N/m. All he models of nonlineariies are similar, i.e. quadraic funcion of splacemen, and parameers of hese funcions were he same, ecep of sign, because proec characer of each nonlineariy. Then, he nonlineariy of hree parameers are described by following equaions: (8) (9) (1) The splacemen is given in meers. All coefficiens in equaions (8 (1) have realisic values. Dependences (8) (1) are drawn in Figs. (13) (15). k () ( ) k( 1 ) ( ) Bl( 1 ) ( ) ( 1 ) k Bl he nonlineariy of siffness, force facor and inducance, respecively. Fig. 16. Inermodulaion sorion caused by nonlinear suspension siffness given by q. (8). ciaion: f 1 5 Hz, U 1 1 V, f 15 Hz, U,5 V Fig. 13. The dependence of suspension siffness on splacemen Bl( ) Fig. 17. Inermodulaion sorion caused by nonlinear force facor Bl given by q. (9). ciaion he same as in Fig. 16. Fig. 14. The dependence of force facor on splacemen e() Fig. 15. The dependence of voice-coil inducance on splacemen The resuls of modeling of inermodulaion sorion wih he same eciaion as described in secion 4 are presened in Figs. 16, 17 and 18 o show he influence of Fig. 18. Inermodulaion sorion caused by nonlinear voice-coil inducance given by q. (1). ciaion he same as in Fig. 16. I can be observed ha he nonlinear inducance causes he smalles harmonic sorion in low frequency range, bu he highes one in high frequency range. The highes inermodulaion sorion is produced by nonlinear Bl, however, his sorion produced by nonlinear inducance is comparable. The inermodulaion sorion produced by siffness nonlineariy is significanly smaller.

6 6 Conclusion The resuls show a grea usefulness of modeling. In an acual loudspeaker he fferen causes of nonlinear sorion occur commonly and i is impossible o separae hem from measuremen resuls. However, he eperimen is necessary. The mehods of measuremens of suspension siffness and Bl-facor has been developed, he mehod of measuremen of inducance nonlineariy is prepared. The nonlineariy of inducance using AC signal as funcion of boh: he curren and he splacemen has been developed. I has been proved ha i is very fficul o conclude abou DC inducance nonlineariy on he basis of hese resuls. I also has been proved ha he nonlineariy of voicecoil inducance is a significan source of sorion, paricularly in high frequency range and for inermodulaion sorions. [1] Voishvillo A., Terekhov A., Czerwinski., Aleandrov S., Graphing, Inerpreaion, and Comparison of Resuls of oudspeaker Nonlinear Disorion Measuremens, J. Auo ng. Soc., vol. 5, No. 4, (4) [11] emarquand G., "Ironless loudspeakers." I Trans. Mag., vol. 43, n 8, pp , 7. [1] Ravaud R., emarquand G., emarquand V., Depollier C., Ironless loudspeakers wih ferrofluid seals, Archives of Acousics, Vol. 33, n 4 (s), pp 53-58, 8. [13] Dobrucki A.: lecroacousic ransducers (in Polish), WNT, Warszawa 7 [14] Cunningham W., Nonlinear Disorion in Dynamic oudspeakers Due o Magneic ffecs, J. Acous. Soc. Am., vol. 1, No. 3, -7 (1949) References [1] Ravaud R., emarquand G., Roussel T.: Timevarying non linear modeling of elecrodynamic loudspeakers, Applied Acousics, 7(9), [] Ravaud R., emarquand G, Roussel T., emarquand V.: Ranking of he nonlineariies of elecrodynamic loudspeakers Archives of Acousics, Vol. 35, n 1, 1. [3] Meri B., emarquand V., emarquand G., Dobrucki A., Moor nonlineariies in elecrodynamic loudspeakers: modeling and measuremen, Archives of Acousics, vol. 34, No. 4, (9) [4] Dobrucki A., Nonypical effecs in an elecrodynamic loudspeaker wih a nonhomogeneous magneic field in he air gap and nonlinear suspensions, J. Auo ng. Soc., vol. 4, No. 7/8, (1994) [5] Aldoshina I., Voishvillo A., Mazin V., oudspeaker Moor Nonlinear Modeling Based on Calculaed Magneic Field in he Gap, J. Auo ng. Soc., vol. 4, No. 1, (1994) [6] Aldoshina I., Voishvillo A., Mazin V., Modeling of Flu Modulaion Disorion in Moving Coil oudspeakers by he Finie lemen Mehod, J. Auo ng. Soc., vol. 43, No. 5, 4 (1995) [7] Voishvillo A., Mazin V., Finie lemen Mehod Modeling of ddy Currens and Their Influence on Nonlinear Disorion in lecrodynamic oudspeakers, J. Auo ng. Soc., vol. 43, No. 1, (1995) [8] Klippel W., Nonlinear arge-signal Behavior of lecrodynamic oudspeakers a ow Frequencies, J. Auo ng. Soc., vol. 4, No. 6, (199) [9] Klippel W., Tuorial: oudspeaker Nonlineariies Causes, Parameers, Sympoms, J. Auo ng. Soc., vol. 54, No. 1, (6)