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Model and estimation method for predicting the sound radiated bya horn loudspeaker – With application to a car horn

Guillaume Lemaitre a,c,*, Boris Letinturier b, Bruno Gazengel c

a IRCAM – Sound Perception and Design team, F-75004 Paris, Franceb SCE-Klaxon, 106 ZI du Clos de la Reine, F-78410 Aubergenville, France

c Laboratoire d’Acoustique, de l’Universite du Maine, F-72000 Le Mans, France

Received 16 December 2005; received in revised form 13 October 2006; accepted 16 October 2006Available online 1 December 2006

Abstract

This paper deals with a new car horn device made of a sound synthesizer and an electrodynamic horn loudspeaker. It presents an one-dimensional model allowing to predict the loudspeaker efficiency and a specific method to estimate experimentally the model parameters.First, this model aims at reducing the time spent in the design process. Second it aims at correcting the sound emitted by the sound syn-thesizer in order that the listener hears the sound designed for creating the warning message. The study gives a survey of the vast loud-speaker literature. It is based on the conventional electroacoustic approach used for electrodynamic loudspeakers and on wavepropagation models used for characterizing acoustic horns. The estimation of the model parameter values is performed using measure-ments of the electrical impedance of the loudspeaker and of the acoustic impedance of the horn. The model is assessed by comparing thecalculated and measured electrical impedances and horn efficiencies. Results show that the model predicts well the horn efficiency up to2500 Hz, the limitation being due to the horn radiation impedance modelization.� 2006 Elsevier Ltd. All rights reserved.

PACS: 43.38.Dv; 43.38.Ja; 43.38.Tj

Keywords: Horn loudspeaker modelization; Parameter estimation method

1. Introduction

Nowadays, automotive industry is paying more andmore attention to the sounds emitted by cars, and espe-cially to the sounds of car horns, which can be a decidingfactor for potential car buyers. The aim of manufacturersis now to design different car horns for different cars, justas they already design many other sounds such as this of

a closing door. Yet car horns are peculiar, since they mustaddress a warning message to the road users [1]. Thus carhorn builders have to design sounds that fulfill the carbuilders wishes, and at the same time they must ensure thatthese sounds are understood as danger signals.

Car horns have been built up to now using electromag-netic excitors coupled together with acoustical (horn) ormechanical resonators (disk). Unfortunately, these technol-ogies do not enable to design new sounds, unless the hornbuilder adapts the resonator. Hence, a new device has beendesigned, made of a synthesizer and of an electrodynamichorn loudspeaker. The electrodynamic loudspeaker hasbeen used for many years to provide audio signals andhas shown its capability in reproducing many differentsounds (see [2] for a review).

0003-682X/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.apacoust.2006.10.005

* Corresponding author. Address: IRCAM – Sound Perception andDesign team, 1 Place Stravinsky, F-75004 Paris, France. Tel./fax: +33 492385083.

E-mail addresses: [email protected] (G. Lemaitre), [email protected] (B. Letinturier), [email protected] (B. Gazengel).

www.elsevier.com/locate/apacoust

Available online at www.sciencedirect.com

Applied Acoustics 69 (2008) 47–59


In previous studies [3–5], we have defined acousticrequirements that allow synthesized sounds to be associ-ated with the warning message. But because of the loud-speaker response, the sounds heard by road-users maybe fairly different from the laboratory fine-designedsounds. Moreover, the horn must meet particular require-ments concerning the acoustic level (105 dB(A) at 2 m).Hence, the loudspeaker behavior has to be integratedalong the design process: sound designers need a physicalmodel of the loudspeaker to anticipate the sound modifi-cations, and loudspeaker designers need a physical modelto make the loudspeaker compatible with the signalrequirements.

The purpose of the study is first to provide with amodel of the horn loudspeaker made of an electrodynamicdriver coupled together with an acoustic horn. A hugeamount of studies dealing whether with electrodynamicalloudspeakers or with horn propagation have been pre-sented. On this basis, we propose a global model of thehorn loudspeaker and we define a parameter estimationmethod. This method is designed to be easily implementedin an industrial context. Finally, we assess this physicalmodel in order to define the frequency range in which itcan be applied.

In Section 2, we present the state of the art of linearphysical models. We also discuss the choice of linear mod-els. In Section 3, we develop the horn loudspeaker modelusing a quadripole formulation. In Section 4 we presentthe method used to estimate the values of the input param-eters of the model. This method is based on impedancemeasurements of the loudspeaker loaded with differentclosed volumes. Then, we characterize experimentally thehorn and we assess the uncertainty in the parameter estima-tion in Section 5. Finally, we discuss the limits of the phys-ical model in Section 6.

2. State of the art

The car horn is made of an electrodynamical drivercoupled together with a compression chamber and an

acoustic horn (see Fig. 1). There are two families of phys-ical model for such systems: the linear and the non-linearmodels. Linear models enable to describe the horn loud-speaker as a four-pole system, the characteristics of whichdepend on frequency. Non-linear models take intoaccount the non-linearity of the driver (see [6–8]) andthe non-linear propagation of sound in the compressionchamber and in the horn (see [9] for a review). In the caseof horn loudspeaker, non-linearities in the horn are themajor cause of distortion [10]. Global non-linear modelsof horn loudspeakers have been proposed in e.g. [11].These models are very complex and thus allow to finelyshape the response of the device by means of feedbackor feed-forward control systems (e.g. in order to reducedistortion). However, they can not lead to simple devicedesign modifications, since they hardly allow to identifywhich mechanical or electrical cause is responsible forthe observed distortions. Moreover, they are hardly usablein an industrial context, because of the careful need ofparameter estimation. For these reasons, we decide touse a linear model of the loudspeaker. The following para-graphs present a state of the art of linear models of loud-speaker and horns.

2.1. Direct radiation loudspeaker models

Most of studies dealing with electrodynamic loud-speaker models are based on the so-called Thiele and

Small’s model. This model results from the work of Thielefirst published in 1961 [12], formalized and popularized tenyears after by Small [13,14]. Some work had already beendone before on the basis of Beranek’s [15] and Olson’s[16] studies, but Thiele and Small’s model had the advanta-ges of both simplicity and accuracy, and shortly became astandard.

This model is a lumped-elements electroacousticalmodel. Each part of the loudspeaker (voice coil, movingelement, diaphragm, etc.) is modelized with an electricalanalogy (resistor, capacitor, etc.). It allows to predict theloudspeaker efficiency (radiated acoustical power as a func-

Compression chamber

Horn

Magnet

Voice coil Phase plug

Diaphragm

Housing

Fig. 1. Cross-section vue of the loudspeaker.

48 G. Lemaitre et al. / Applied Acoustics 69 (2008) 47–59


tion of electrical power)1 and hence to assess the influenceof each component.

Although this model is defined by a few parameters, theexperimental accurate estimation of these parameters maybe difficult. For a long time, parameters have been esti-mated from loudspeaker frequency response properties(resonance frequencies, damping factor, etc.) [17]. Sinceearly 80s, however improvements of computers have madeglobal model fitting techniques possible. The basic princi-ple is to run an error minimization algorithm between themeasured and the predicted response. The extreme case isto model the loudspeaker as a two-port ‘‘black box’’.Experimental procedures and curve fitting techniquesamounts then in measuring the elements of the two-portmodel, without any link to the physical constitutive ele-ments of the loudspeaker. Such methods are of very com-mon use for non-linear models (see for example [18]), andhave also been used for linear models [19–21]. Their predic-tions are much more accurate, but they can not lead todesign specifications.

Thiele and Small’s model is based on some physical sim-plifications (lumped element hypothesis) that let the modelfail to predict fine and high frequency phenomena. Furtherwork has aimed at improving the fine modelization of eachelement. The influence of the diaphragm shape and of theresonance modes have been studied in, e.g. [22]. Electricalphenomena within the voice coil (such as Eddy current)have been studied and modelized in [23]. Suspension mod-elization was improved by Knudsen and Jensen [24].

There exists some other studies that have been devel-oped parallel to Thiele and Small’s model [25]. They pro-pose much more precise and physically grounded modelsof the mechanical part of the loudspeaker. But these mod-els can no longer be described by lumped-element circuits.Their predictions are much accurate than Thiele andSmall’s models, but their complexity, and the very delicateparameter estimation protocols made them very difficult touse in an industrial application.

2.2. Horn loudspeakers

The propagation of acoustic waves in horns have beenprincipally described in musical acoustics studies. Acousti-cal propagation in horn is described in Nederveen’s [26]reference book. The formulation used in most of electro-acoustical studies is summarized in [16].

2.2.1. Acoustic propagation in waveguides

The fundamental equation describing wave propagationin variable section waveguides is the Webster equation

[27,28]. It allows to express the relationship between input(throat of the horn) and output (mouth of the horn) quan-tities: acoustic pressure and volume velocity. Hence one

boundary condition (the radiating impedance at hornmouth) is required for solving the system. Computing radi-ation impedance in the general case is a very complex prob-lem. Levine and Schwinger [29] give a complex analyticalexpression for a cylindrical pipe radiating in an infinitespace. Useful approximations are given by Causse et al.[30], Dalmont et al. [31] (based on Norris and Sheng’s work[32]), and Helie and Rodet [33].

Dealing with the wave propagation, several approachesare available. First, a horn can be described in the fre-quency domain by its throat impedance (or equivalentlyby transmission matrices). An usual assumption is thatwaves propagating along the horns are either plane orspherical, but Benade and Jansson [34,35] showed thatbecause of axial to radial mode conversion, none of thishypotheses is fairly true. However, a fruitful approachhas been introduced by Plitnik and Strong [36]. It consistsin numerizing any bore geometry by a series of cylinders.Causse et al. [30] further generalized this method to includeviscothermal losses. A same approach is used by Hollandet al. [37] for the loudspeaker horn. They numerize thehorn as a series of exponential elements, taking axial toradial mode conversion into account. Another approachis to modelize the horn with an electroacoustical circuit[38]. Finally, wave propagation can be described in the timedomain. This approach has been used for real-time simula-tion of musical instruments (see e.g. [39]).

2.2.2. Loudspeaker horns

Loudspeaker horns allow to adapt impedance betweenthe diaphragm and the propagation medium. They forcethe diaphragm movements to remain small, and they max-imize the amount of energy radiated to the medium [40,41].Attempts to optimize energy transmission has lead to newgeometries of horns [42].

Horns have a strong directivity, that may greatly varieswith frequency. Conical horns are known to be the simplestbore geometry allowing an uniform directivity [43]. But thishypothesis is valid only below a particular frequency(‘‘break wave number’’) [42]. Several horn geometries havethen been proposed to control directivity over a large fre-quency range [44].

Horns are most of the times coupled together with theloudspeaker diaphragm by means of an element calledthe compression chamber. This element aims at adaptatingimpedance between a large diaphragm radius and a smallerhorn throat radius. Its specific inside geometry (the phase

plug) is designed to cancel radial modes at throat [45],which would under-optimize energy transmission into thehorn. An electroacoustical model has been proposed byHenricksen [46].

Finally, only a few studies has been published for pro-posing a global model of a horn loudspeaker. Leach [47]proposed a global electroacoustical model of an exponen-tial horn loudspeaker, and demonstrated its band-pass fil-ter behavior. Geddes and Clark [48] published a rathercomplex global model, with a specific resolution of the

1 Note that this definition of efficiency is not this one proposed by Thiele[12]. It rather corresponds to sensitivity, in the case of a microphone.

G. Lemaitre et al. / Applied Acoustics 69 (2008) 47–59 49


Webster equation. Their predictions fitted measures verywell.

3. Horn loudspeaker model

3.1. Description of the loudspeaker

The loudspeaker under study is made of three parts: anelectrical to mechanical transducing system (the voice coilwithin the magnetic field), a mechanical part (the movingelement, made of the suspended diaphragm and coil), andan acoustical load (compression chamber and horn). Thetwo former parts define the electrodynamic driver, whilethe latter part defines the resonator (see Fig. 1).

3.2. General formulation of the problem

Assuming a 1-D propagation model, the horn can berepresented by an equivalent electrical circuit shown inFig. 2 (see Tables A.1 and A.2 in Appendix A for a glossaryof symbols). Formally the problem is defined by four inde-pendant variables (ug, ig, pd, qd), which represent respec-tively the input voltage and current, the acoustic pressurein front of the diaphragm and the volume velocity gener-ated by the diaphragm. They are linked together by

ug

ig

� �¼

Sd

Bl ZecBlSd

1þ Zec

Zem

� �Sd

BlBlSd

1Zem

24

35 pd

qd

� �; ð1Þ

where Sd is the effective surface of the diaphragm, Bl theforce factor, Zec the electrical voice coil impedance,Zem = Bl2/Zmm the motional impedance (written in theelectrical analogy), and Zmm is the motional impedancewritten in the mechanical analogy. The relation betweenthe acoustic pressure pd and acoustic volume velocity qd

is given by

pd

qd

¼ Zal; ð2Þ

where Zal is the acoustical load impedance which repre-sents the boundary condition. The pressure radiated athorn mouth pm is given by

pm

pd

¼ T load; ð3Þ

where Tload is the transmission matrix of the horn.The problem is completly defined by the three equations

(1)–(3). The following paragraphs develop the formulationof Zec, Zmm, Zal, and Tload corresponding to the electrical,mechanical and acoustical parts of the loudspeakers.

3.3. Voice coil, moving element, and acoustical load

3.3.1. Voice coil

The voice coil is made of a copper winding. The simplestmodel is a resistance and an inductance connected in series.But many empirical studies have shown that this modelfails for high frequencies [12,49,50,23,51]. Vanderkooyhas explained this phenomenon by the presence of Eddycurrents at the magnet surface [51], and proposed a modelto account for these currents (Re + Kx1/2 + jKx1/2 imped-ance behavior). Wright [50] has then proposed to generalizethis model, since it was not able to explain some of thereported phenomena at high frequencies, on the basis ofempirical measurements (Re þ KrxX r þ jKixX i impedancebehavior). We adopt a similar empirical model derived tofit our own measurements of the voice coil impedanceout of the magnetic field

Zec ¼ Recð1þ x=xrÞr þ jðxLecÞa; ð4Þwhere xr, r, and a are empirical parameters which accountfor the increase of the resistance and of the decrease of theinductance at high frequencies (note that the asymptoticbehavior of this model is identical to Wright’s).

3.3.2. Moving element

The dome-like loudspeaker diaphragm is suspended tothe loudspeaker housing by peripherical corrugations ofthe membrane. Assuming that the membrane acts like arigid piston, this moving element can be modelized by asingle mechanical mass Mmm linked to a mechanical com-pliance Cmm with mechanical losses Rmm (Kelvin’s modelof suspension [24]). The motional impedance in the electri-cal domain is then

Bl:1 Sd:1

Ug

ig

Zec Zmmvd qd

Zal

Fig. 2. Equivalent electroacoustical circuit of the loudspeaker.



Zem ¼ Rec

Qm

Qe

j xx0

Q�1m

1þ j xx0

Q�1m þ ðj x

x0Þ2; ð5Þ

where

x0 ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

MmmCmm

p ; ð6aÞ

Qm ¼1

RmmCmmx0

; ð6bÞ

Qe ¼QmRec

Bl2=Rmm

; ð6cÞ

are, respectively, the diaphragm resonance frequency, themechanical quality factor, and the electrical quality factor.

3.3.3. Acoustical impedance

3.3.3.1. Compression chamber and phase plug. Althoughthere exists some electroacoustical models in the literature[46,48,53], we propose a semi-empirical model of the com-pression chamber. It is based both on Smith’s analyticalsolution [45] and on geometrical simplifications (seeFig. 3). First, a thin air layer (volume Vc0) is compressedbetween the diaphragm (section Sd) and the phase plug.We modelize it as an acoustical compliance Cac0 = Vc0/qc2. Then waves are propagated along the slits locatedbetween the phase plug and the compression chamber walls(annular section Sa): we simplify it in a cylindrical wave-guide2 of section St and of length Lc1. Finally, waves prop-agate to the horn throat via a small cylindrical pipe (sectionSt, radius rt).

With this approximation, and after Causse et al. [30], theinput admittance Yal of the compression chamber loadedby an acoustical impedance Zat is given by

Y al ¼1

Zal

¼ jCac0xþtanh½jkLc1 þ a tanhðft=ZatÞ�

ft

; ð7Þ

where k is the wave number including viscothermal lossesat the walls, and ft is the characteristic impedance at thethroat. After [54,30], they can be approximated at a tem-perature of 20 �C by

k ¼ xc

1þ 3� 10�5 1� j

2prt

ffiffiffifp � 3� 10�8 j

2pr2t f

� �; ð8aÞ

ft ¼qcSt

1þ 7:61� 10�4

rt

ffiffiffifp � j

4:89� 10�6

r2t f

� �; ð8bÞ

where c is the sound speed and q is the air density at rest.The pressure at the compression chamber output (hornthroat) pt is related to the pressure at the diaphragm pd by

pt

pd

¼ T pipeðLc1; rt; ZatÞ ¼1

cosðkLc1Þ þ jft=Zat sinðkLc1Þ: ð9Þ

3.3.3.2. Horn. As the cone angles are small, we assumeplane wave propagation (for a 20� angle, plane and spher-ical wave front area are only 1% different). Hence, after[30], acoustical throat impedance Zat of a conical hornloaded by an acoustical mouth radiation impedance Zar is

Zat ¼ft

1jkxtþ tanh jkLþ a tanh fm

Zar� 1

jkxm

� �h i : ð10Þ

Pressure at cone mouth Pm is given as a function of throatpressure Pt by

P m

P t

¼ T hornðxt; xm; L; ZarÞ

¼ xt

xm

1

cosðkLÞ þ j sinðkLÞ fm

Zar� 1

jkxm

� � ; ð11Þ

where xt, xm, and L are given in Fig. 4 and are reminded inTable A.2. Wave number k and characteristic impedancesft and fm are given in Eq. (8).

3.3.3.3. Radiation impedance. Radiation impedance modelscan be found in three studies [30,31,33] dealing with unbaf-fled radiating waveguides. We use the Dalmont et al. [31]approximation of Levine and Schwinger’s [29] and Norrisand Sheng’s work [32]. Hence the radiation impedanceZar of an oscillating piston of radius rm and section Sm is

Zar ¼qcSm

1� jR0je�2jkd0

1þ jR0je�2jkd0; ð12Þ

Sd

St

St

Lc1

Sd

Vc1

Vc0

Fig. 3. Geometrical approximation for the compression chamber and thephase plug.

2 As there are no radial modes, we assume plane wave propagation.

xt

xm

L

St Smapex

throat

mouth

Fig. 4. Conical horn dimensions.



where

jR0j ¼1þ 0:2krm � 0:084ðkrmÞ2

1þ 0:2krm þ ð0:416ðkrmÞ2Þ; ð13aÞ

d0 ¼ 0:6133rm1þ 0:44ðkrmÞ2

1þ 0:19ðkrmÞ2� 0:02 sin2ð2krmÞ

" #; ð13bÞ

and k is defined in Eq. (8).

3.4. Electrical impedance and horn efficiency

Finally, we can derive the electrical impedance and theefficiency of the loudspeaker from the above formulations.From Eq. (1), the loudspeaker electrical input impedanceZhp is

Zhp ¼ Zec þ ZeM; ð14Þwhere Zec is the electrical impedance of the voice coil (seeEq. (4)) and ZeM = (ZelZem)/(Zel + Zem) is the impedanceof the mechano-acoustical load expressed in the electricaldomain. ZeM depends on Zem (Eq. (5)) and onZel ¼ Bl=ðS2

dZalÞ (see Eqs. (7), (10), and (12)).Combining Eqs. (1), (9), and (11) allows to derive the

loudspeaker efficiency

pm

ug

¼ Zal

Sd

Bl

� ZeM

Zec þ ZeM

T pipeðLc1; rt; ZtÞT hornðxt; xm; L; ZarÞ; ð15Þ

where the total acoustic load impedance Zal is defined byEqs. (7), (10), and (12), the coil impedance Zec is definedby Eq. (4), the pipe and horn transmission functions Tpipe-(Lc1, rt, Zg) and Thorn(xt, xm, L, Zar) are defined by Eqs. (9)and (11).

4. Parameter estimation method

The loudspeaker model presented in Section 3 is definedby 14 parameters: the coupling factors ratio Sd/Bl, thevoice coil parameters (Rec, Lec, xr, r, and a), the movingelement parameters (x0, Qm, and Qe) and the acoustic loadparameters (Vc0, Lc1, rt, rm, and L).

The usual methods used to estimate the driver parame-ters imply to measure the electrical impedance when thediaphragm is freely radiating. This not possible in our case,since the diaphragm is fixed to the housing by the compres-sion chamber. We have measured that unmounting thecompression chamber (to let the diaphragm freely radiat-ing), and then remounting it changes dramatically themechanical parameter values. Thus our method consistsfirst in measuring the horn and compression chamber inputacoustical impedances for the compression chamber andthe horn alone (this can be done once and for once, becausetheir dimension do not vary from one loudspeaker toanother). Then, without unmounting the compressionchamber, we add different acoustical loads to the compres-

sion chambers, and we compare the different measurementsof the electrical impedance. This method is, in spirit, closedto those developed by Makarski and Behler [20,21] for theirtwo-port ‘‘black-box model’’. These authors describe amodel of loudspeaker similar to ours (described by Eqs.(1)–(3)). The difference is that the four elements of thematrix in Eq. (1) are not analytically related to the consti-tutive elements of the loudspeaker (e.g. mass of the movingelement, length of wire, etc.), but rather written asunknown frequency-dependent functions, that have to bedescribed numerically by measurements. By means of origi-nal measurement methods (acoustical impedance of aKundt’s tube loaded by the loudspeaker driver, electricalimpedance of two coupled drivers, etc.) and boundary ele-ment simulations of the horn, they are able to measurenumerically the four functions of the matrix. The testsshow that this method is able to provide accurate predic-tions of the loudspeaker efficiency, for a wide variety ofloudspeakers and horns, and for frequencies up to20 kHz. However, since this model cannot lead to identifyany lumped-element values, it cannot be used to modify thedesign of the loudspeaker constitutive elements.

4.1. Acoustical load parameters

The horn dimensions (rt, rm, and L) are estimated bygeometrical means. To estimate the compression chamberparameters (Vc0, Lc1), we measure the acoustical inputadmittance of the compression chamber closed by a smallpipe of length Lp, and we fit this measure to Eq. (B.1)(Appendix B).

4.2. Coupling factors ratio

Following Le Roux’s work [55,56], the ratio Sd/Bl isestimated by measuring the loudspeaker electrical imped-ance, in two cases. In the first case, the loudspeaker isloaded by the closed compression chamber the equivalentvolume of which is Veq. In the second case, the compressionchamber is closed by a cylindrical pipe of length Lp, sectionSt which leads to an equivalent volumeV p

eq. In both cases(as long as the moving element resonance frequency is farbelow the compression chamber first resonance frequency),we estimate the resonance parameters (resonance frequencyx0 and quality factor Qe), as well as the voice coil resistanceRe. From this estimations, we derive the coupling factorratio Sd/Bl with

BlSd

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqc2

LpLc

Lp � Lcð ÞV p

eq

V eq V eq þ V peq

� �vuut ; ð16Þ

where Lc and Lp are the equivalent mechanical masses3 andare estimated using

3 Superscripted index c means ‘‘loaded with closed compressionchamber’’.



Lc ¼ Rec

xc0Qc

e

; ð17aÞ

Lp ¼ Rec

xp0Qp

e

: ð17bÞ

Eq. (17) are drawn from the inversion of Eq. (B.4) (Appen-dix B).

4.3. Driver parameters

We estimate the driver parameters by measuring theinput electrical impedance when the driver is loaded bythe closed compression chamber. In this case, by fitting thismeasure to Eq. (B.2) (Appendix B), we estimate the closedchamber driver parameters (Qc

m, Qce, xc

0, xr, r, Lec, and a).Finally the free-radiating driver parameters are evaluatedby

Qe ¼ Qce; ð18aÞ

x0 ¼ xc0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Rec

Qexc0

S2d

ðBlÞ2qc2

V eq

s; ð18bÞ

Qm ¼ Qcm

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� Rec

Qexc0

S2d

ðBlÞ2qc2

V eq

s: ð18cÞ

5. Model assessment

In this section, we apply the model and the parameterestimation method described above to a loudspeaker usedas a new prototype of car horn. This allows us to validatethe modelization of each element of the global model, and

to evaluate the estimation method uncertainty. This loud-speaker has a membrane with a 5 cm diameter, and a22 cm length horn (throat diameter: 2 cm, mouth diameter:10 cm). The total weight of the device is 600 g.

5.1. Measurement settings

Parameter estimation requires both electrical and acous-tical impedance measurements. All electrical impedancesare measured with a 0.125 X resistor connected in serieswith the loudspeaker. Sweep measures are made with aStanford Research SR785 analyzer. The loudspeaker isplaced in an anechoic chamber. Electrical impedance mea-surements are compensated for wire transfert function.Accuracy is estimated at ±0.8% for impedance modulus,and ±0.1� for impedance phase. During measurements ofboth electrical impedance and loudspeaker efficiency, theloudspeaker is fed with a 10 mV input voltage, whichensures that the average total harmonic distortion is lessthan 1%. Acoustical impedances are measured with animpedance head described in [57,58]. Loudspeaker effi-ciency is measured by recording simultaneously pressureradiated at horn mouth and loudspeaker input voltage.Pressure is measured with a Sennheiser KE4 electret micro-phone (sensibility 153 mV/Pa).

5.2. Experimental determination of parameters

5.2.1. Compression chamber parameters

To estimate Vc0 and Lc1, we measure the input acousti-cal impedance of the compression chamber closed by a

102

103

80

90

100

110

120

130

140

150

Frequency in Hz

Mod

ulus

in d

B

Measure

Fitted model

Fig. 5. Measured and predicted input acoustical impedance of the compression chamber loaded with a 60 mm long closed pipe.



60 mm long cylindrical pipe. The length of the pipe let thesystem resonance frequencies fall within the sensor band-width. A minimum least square fitting method is appliedwith 500 points between 100 Hz and 10 kHz (see Fig. 5).This figure shows that the two first measured resonance fre-quencies (sensor bandwidth) correspond to the predictedfrequencies with an error smaller than the sensor accuracy,which validates our model. But the measured amplitudesare about 6 dB lower than the predicted ones. This error,greater than sensor accuracy, may be interpreted as visco-thermal losses underestimation. Below 400 Hz, the mea-sured slope is steeper than the predicted compliancebehavior (�20 dB/decade). This may be due either to sys-tematic failure of the measuring device at low frequencies,or to a phenomenon which is not taken into account in themodel.

According to the fitting method confidence interval, thismethod allows to estimate Vc0 with a precision of ±1.7%and Lc1 with a precision of ±1.1%.

5.2.2. Horn parameters

First, the input acoustical impedance of the closed hornis studied in order to test the horn model without takinginto account the radiation impedance. Within the sensorbandwidth, the differences between the estimated and themeasured resonance frequencies are less than the sensoraccuracy. The estimated resonance amplitude are 5 dBhigher than the measured ones, which suggests that visco-thermal losses are underestimated (see top panel ofFig. 6). Second, the impedance of the freely radiatinghorn is measured. Bottom panel of Fig. 6 shows that the

model predictions and the measurements are consistantfor the frequency domain [100 Hz to 2 kHz] indicating thatthe radiation impedance model is valid in this frequencyrange.

5.2.3. Coupling factor ratioSd/Bl is estimated by measuring the electrical impedance

of the loudspeaker loaded with the closed compressionchamber, and afterwards with the compression chamberterminated by a closed pipe (27.6 mm). Eq. (B.2) and mea-surements are fitted with Akabak software [52]. In order toevaluate the reproducibility errors, estimations arerepeated 10 times. The standard deviation of the reproduc-ibility error is ±1.6%.

5.2.4. Driver parameters

To estimate the driver parameters, we measure the elec-trical impedance of the loudspeaker with the closed com-pression chamber and fit it to Eq. (B.2) with Akabaksoftware [52] (see Fig. 7). It can be noted that the compres-sion chamber first resonance appears around 3.5 kHz. Werepeat the estimation 10 times. The reproduciblity errorsare listed in Table 1.

This shows that the mechanical quality factor Qcm is the

most inaccurate value. This explains why Eq. (16) is basedonly on Qe. The voice coil parameters fr and Leb estima-tions are also less accurate. Their estimation is biased bycompression chamber modes. When we propagate thiserror to the estimation of the parameters for the freely radi-ating driver, we found that the most inacurrately estimatedparameter is Cmm: ±6.0%.

df

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000100

110

120

130

140

Mod

ulus

in d

B

Frequency in Hz

Measure

Fitted model

Freely radiating horn

0 1000 2000 3000 4000 5000 6000 7000 8000 900080

100

120

140

160

Mod

ulus

in d

B

Closed horn

Fig. 6. Acoustical input impedance of the closed and of the freely radiating horn.



5.3. Global model evaluation

We estimate the validity of the physical model developedin Section 3 by comparing the predicted and the measuredelectrical loudspeaker input impedances, and the predictedand the measured loudspeaker efficiencies (radiated pres-sure over input voltage). Measured and estimated efficien-cies are reported in Fig. 8. For frequencies below2.5 kHz, error between measured and predicted efficiency

102

103

104

0

20

40

60

80

100

120

Mod

ulus

in O

hms

102

103

104

-1.5

-1

-0.5

0

0.5

1

1.5

Frequency in Hz

Pha

se in

rad

ians

Measure

Fitted model

Fig. 7. Electrical impedance of the loudspeaker loaded with the closed compression chamber.

Table 1Estimated reproducibility error of driver parameters, for the driver beingloaded by the closed compression chamber

Parameter Standard deviationa

(%)Parameter Standard deviationa

(%)

Qcm 4.2 Rc

eb 0.2f c

r 2.0 Qce 0.2

Lceb 1.4 f c

0 0.1ac 0.8 rc 0.0

a % of average value.

102

103

104

-20

-10

0

10

20

30

Mod

ulus

in d

B (

Pa/

V)

102

103

104

-80

-60

-40

-20

0

Frequency in Hz

Pha

se in

rad

ians

ModelMeasure

Fig. 8. Efficiency (mouth pressure over input voltage) of the free radiating loudspeaker.



is less than 2 dB. For frequencies between 2.5 kHz and5 kHz, error is about 10 dB. Around 5 kHz and 7 kHzthe measured response dramatically drops from the pre-dicted one. Hence the predictions of the model are usablefor frequencies below 2.5 kHz. For frequencies above2.5 kHz, the radiation impedance model fails.

6. Discussion

In this paper, we have proposed a model using 14parameters for describing the horn loudspeaker made ofan electrodynamic driver coupled together with a compres-sion chamber and a horn. The model is globally defined bya quadripole equation and a boundary condition. This for-mulation allows to easily express any desired quantity. Wehave defined a method that allows to estimate these param-eters without unmounting the compression chamber, whichwould change the mechanical characteristics of the loud-speaker. The method is based on electrical impedance mea-surements of the loudspeaker loaded by different acousticvolumes. It needs to estimate the compression chamberequivalent volume when the driver is unmounted.

We have applied this model and the estimation methodto a loudspeaker used as a new car horn. Due to confiden-tiality restrictions, we are not allowed to disclose here thenumerical values of the estimated parameters. However,the modelization of each element fits rather well with themeasured quantities. The estimation reproducibility errorsare estimated of a few percents.

This model, and the associated parameter estimationmethod, suffers however from several drawbacks. Electricaland mechanical driver parameters have to be estimatedwith a curve fitting technique based on a measure of electri-cal impedance. To let the fitting algorithm converge, weneed to simplify the models. While the loudspeaker behav-ior is fairly predicted by these simplified models for low fre-quencies, higher frequency response is clearly, although ona small amount, influenced by the compression chamberresonance modes. Our model is theoretically able to takethis effect into account. This would however result in anover-parametrized model, that would make the curve fit-ting algorithm unable to converge.

The acoustic models of both compression chamber andconical horn used in this study fit the experimental dataonly for a limited bandwith up to 3.5 kHz, and so is limitedthe prediction ability of the global model. Within thisbandwidth, the model predictions are closed to measure-ments. More precise models are available for compressionchamber [59] and for horn (chaining small cylindrical ele-ments to fit the precise bore geometry and better estimateviscothermal losses [36]), but the main limitation resultsfrom the radiation impedance model. A solution wouldbe to generalize Helie’s work [33] to our cone dimensions.

Then, this model allows to predict the loudspeaker lin-ear behavior. Globally, this device acts like a band-pass fil-ter. Its bandwith is approximatively 500 Hz to 2 kHz. Thisis rather narrow, since our previous work has recom-

mended that car horns fundamental frequency must liearound 400 Hz, and sounds have to have a rich spectralcontent. Hence, loudspeaker design has to be reviewed toincrease bandwith. Our model allows to predict how somedevice modifications may help to reach this goal. Thismodel also allows to inverse-filter signal to compensatefor loudspeaker response at high frequencies. This is, how-ever, not possible for low frequencies, because energy boostat low frequencies would dramatically increase distortion.Finally, this model allows to listen to the sounds thatwould really be emitted by the device, and not only the lab-oratory created sounds. Therefore, signal specifications canbe tested directly on the sound that would be heard by roadusers.

This model, however, is a linear model that permits topredict signal linear filtering, but does not allow to predictharmonic and intermodulation distortions. As car horn sig-nals are very loud signals, it can be assumed that non-linearphenomena occur when the loudspeaker is used as a carhorn. Then, the next step will be to include the mainnon-linear phenomena in the loudspeaker model. However,care must be taken to design a non-linear model that wouldbe easily related to physical parameters.

7. Conclusion

In the context of sound quality, car horn builders areattempting to create new sounds by means of synthesizersand electrodynamic horn loudspeakers. A previous study[3] has provided some sound design specifications, in orderto create new sounds which are still perceived as warningmessages. However, as the laboratory-designed sounds willbe altered by the loudspeaker, the loudspeaker responsemust be included in the design process.

Based on the huge literature dealing with loudspeaker,and on both theoretical and empirical considerations, wehave proposed a linear model with 14 parameters for thishorn loudspeaker. The problem is formalized by a matrixequation and a boundary condition. To explicit each termof this system, we have further modelized impedances ofthe different loudspeaker elements, electrical, mechanicaland acoustical. Our model is mainly based on Thiele andSmall’s model, geometrical modelization of the compres-sion chamber and acoustical studies of wave propagationin horns. We have also designed a parameter estimationtechnique, which allows to estimate the model parametervalues with methods that are easily implementable in anindustrial context. Within our loudspeaker bandwith(500 Hz to 2 kHz), this model is able to fairly predict theloudspeaker behavior.

Acknowledgements

This study has been founded by SCE Klaxon and by thefrench ministry of industry. The authors thank Jean-PierreDalmont at the Laboratoire d’Acoustique de l’Universitedu Maine for fruitful discussions and for his help in mea-



suring acoustical impedances, and two anonymous review-ers for their comments on previous versions of this article.

Appendix A. A symbols notation

Table A.1 and Table A.2

Appendix B. Loudspeaker loaded with a small cylindrical

pipe

B.1. Acoustical impedance

When the compression chamber is loaded by a short cyl-indrial pipe of same section St and length Lp, Eq. (7)becomes4

Y pal ¼ jCac0xþ

1

ft

j tan kðLc1 þ LpÞ� �

; ðB:1Þ

where Cac0 = Vc0/qc is the acoustical compliance equiva-lent to volume Vc0. The input acoustical impedance ofthe closed-pipe loaded compression chamber exhibits thenmaxima and minima (resonance) for frequenciesxn ¼ ð2nþ 1Þ pc

2ðLc1þLpÞ. Below the first resonance frequency,compression chamber behaves like a pure acoustical com-pliance, equivalent to an air volume Veq = Vc0 + Vc1 +StLp.

B.2. Loudspeaker electrical impedance

Below compression chamber first resonance frequency,loudspeaker electric input impedance may be approxi-mated by Eq. (B.2)

Zphp ¼ Rec

Qpm

Qpe

j xxp

0

Qpm�1

1þ j xxp

0

Qpm�1 þ j x

xp0

� �2

þ Recð1þ x=xrÞr þ jðxLecÞl ðB:2Þ

with

Rp ¼ Bl2

Rmm

; ðB:3aÞ

Cp ¼ Mmm

Bl2; ðB:3bÞ

1

Lp ¼1

Bl2Cmm

þ S2d

Bl2Ceq

; ðB:3cÞ

and

xp0 ¼

1ffiffiffiffiffiffiffiffiffiffiLpCpp ; ðB:4aÞ

Qpm ¼ x0RpCp; ðB:4bÞ

Qpe ¼ x0RebCp; ðB:4cÞ

Table A.1Glossary of symbols

Symbol Analogy

q – Air densityc – Sound velocityB – Air-gap flux densityl – Length of voice coil wireSd – Effective area of the diaphragmug – Generator voltageig – Voice coil currentvd – Diaphragm velocityqd – Volume velocity in front of diaphragmpd – Pressure at acoustical loadpt – Pressure at horn throatpm – Pressure at horn mouthZec Elec. Voice-coil impedanceRec Elec. Voice-coil resistanceLec Elec. Voice-coil inductancexr – Empirical parameters tor – characterize voice-coila – impedanceZmm Mech. Motional impedanceZem Elec. Motional impedanceZeM Elec. Motional impedance including acoustical loadRmm Mech. Mechanical lossesCmm Mech. Suspensions complianceMmm Mech. Moving element massx0 – Resonance pulsation of the moving elementQm – Quality ratio of the moving elementZal Ac. Input impedance of the acoustical loadYal Ac. Input admittance of the acoustical loadZel Elec. Input impedance of the acoustical loadCal0 Ac. Compliance equivalent to an air volume Veq0

Zat Ac. Horn throat impedanceZhp Elec. Loudspeaker input impedanceZar Ac. Radiation impedanceCac0 Ac. Compliance of air volume Vc0

ft Ac. Charactristic impedance at throatTpipe Ac. Transmission function in a pipeThorn Ac. Transmission function in the horn

Table A.2Glossary of geometrical symbols

Symbol

Sd Diaphragm projected sectionVc Compression chamber equivalent volumeVc0 Volume equivalent to the fluid layer between diaphragm

and phase plugVc1 Volume equivalent to the volume between phase plug and

compression chamber wallsVeq Volume equivalent to the compression chamberV p

eq Volume equivalent to the compression chamber loadedwith a pipe

Lc1 Length of cylindrical pipe of section Sg and volumeequivalent to Vc1

Lp Length of loading pipeSp Section of loading pipeVp Volume of loading pipert Radius of horn throatSt Section at horn throat (or compression chamber mouth)rm Radius at horn mouthSm Section at horn mouthxt, xm, L Horn dimensions

4 Superscript index p means ‘‘loaded with a closed pipe’’.



Rp, Cp, and Lp are, respectively, mechanical equivalentlosses, compliance and mass.

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Author's personal copy - IRCAMarticles.ircam.fr/textes/Lemaitre08a/index.pdfAuthor's personal copy Model and estimation method for predicting the sound radiated by a horn loudspeaker