October 2005

Nowadays, a huge amount of electroacoustic transducers are used forthe reproduction of sound, both of the human voice and of musicalinstruments. Typically we call these transducers loudspeakers. Theseelectroacoustic transducers are one of the world’s most

manufactured technical devices. For example, in the year 2004 nearly 700million cell phones where sold worldwide. Every cell phone has at least onetransducer included. So the cell phone industry alone needs approximately abillion speakers per year.

It should not therefore come as a surprise that there is a lot of serious engineeringwork going on in the design of loudspeakers to develop optimal products – bothin terms of acoustic performance as well as manufacturing costs. However, thefocus on the development work is influenced by the viewpoint of an electricalengineer. This seems to be due to the fact that every loudspeaker is driven bysome kind of electrical device (i.e. an amplifier). But a loudspeaker consists notonly of an electrical device - it is built up of a motor system and a vibrationsystem, which typically converts an electrical signal to a mechanical force insidethe motor system (also known as voice coil), which in turn excites a vibrationsystem consisting at least of a diaphragm with some kind of support. Thediaphragm itself excites the surrounding air to radiate sound. That is, in verysimple words, the functional principle of a loudspeaker.

If we take a closer look at the physical domains involved, we figure out thefollowing fields that a loudspeaker uses:

� electromagnetics

� structural dynamics

� sound radiation

Thus this is often referred to as multiphysics. It is interesting to note that only oneof these domains is a typical electrical domain. A recent panel discussion [1] gavea good and more general overview about the physical domains involved in astimulus to the human ear (see Figure 1). However, in this article we willconcentrate on the mechanical and acoustical domains.

Though different types of loudspeaker exist for different specific applications,these three physical domains are almost always involved. In general the mass ofelectroacoustic transducers can be classified as being of one of the followingtypes:

� woofers

� tweeters

� horns

� small transducers

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Acoustic Matrix Methods foand Small Transducers

Figure 1: Physical domains invloved in the path of a stimulus to the human ear

Figure 2: Example of an electromechanical equivalent circuit

Figure 3: Applicable frequency range for lumped parameter models

Alfred J. Svobodnik,

“In the last few decades, a lot of effort hasbeen put into improving the functionalbehavior of loudspeakers, especially bymeans of numerical simulation”

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We will use this classification here from a physicist’s viewpoint, though audioengineers would indicate that this is not a complete list. But for the purposes ofthis article, outlining the different parts of the physical domains and how theyinteract with each other in different types of loudspeakers, this classification issufficient.

In the last few decades, a lot of effort has been put into improving the functionalbehavior of loudspeakers, especially by means of numerical simulations. Thefocus was, and often enough still is, on improving the electromagnetic behavior(i.e. the motor system) of loudspeakers. Electromagnetic simulations by means offinite or boundary elements were the first matrix methods introduced in speakerdevelopment. However, after having reached a nearly perfect motor system itturned out that the whole system, i.e. the loudspeaker, is far from perfect. Someengineers have concluded that this is due to the fact that they have onlyoptimized only one link of the chain in depth using advanced simulation tools.

Systems optimization must also include the treatment of the structural dynamicbehavior of the vibration system and sound radiation behavior by means ofadvanced simulation models. Thus we are faced with more of a mechanicalsystem than an electrical system. However, the fact is that it is a multiphysicalsystem, and should be treated as such in the search of optimum.

Traditional “Lumped Parameter Models”There is a long tradition of treating a loudspeaker as a multiphysics device byelectrical engineers. Back in the 1950’s, Olson developed a lumped parametermodel which describes the interaction of an electrical system (motor system) witha mechanical system (vibration system) by means of electrical analogies. Itshould be noted that the mechanical system was already treated as beingnonlinear. An example of an electromechanical equivalent circuit is given inFigure 2.

In the early seventies several technical papers were presented to the AES [2] thatresulted in a model which electrical engineers know today as the famous “Thiele-Small Parameters”. A.N. Thiele and R.H. Small developed parameters definingthe relationship between a loudspeaker and a particular enclosure. Over timethere have been a lot of developments to extend these early lumped parametermodels to more sophisticated models. In particular, the recent work of W. Klippelon the large signal behavior, including a lot of nonlinearities in the differentphysical domains, and the identification of parameters via measurements shouldbe mentioned here.

However, all these models have one significant drawback - they are based onlumped parameters which use only a few (or at least one) parameters to describea physical domain. For example the mechanical system, the vibration system, isdescribed by stiffness and mass. From the mechanical engineer’s viewpoint, thevibration system is discretized by a single mass system. This leads to the fact thatthis model is only capable of describing the piston mode of a vibration system.This will lead to limitations in the frequency range where the lumped parametermodel is applicable (see Figure 3).

This fact has led to the development of distributed models based on matrixmethods, where the domains are modeled via systems consisting of manydegrees of freedom.

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or Woofers, Tweeters, Horns

Figure 4: Governing equations describing the structural dynamic behavior of avibration system

Figure 5: Non-pistonic modes of loudspeaker diaphragms (also known as cones)

Figure 6: Helmholtz equation solved via boundary elements

NAD - Numerical Analysis and Design GmbH & Co KG

October 20058

Matrix MethodsAt a first glance there seems to be only a small differencein the governing equations describing for example thevibration system (see Figure 4) by matrix methods and bylumped parameter models. But the big difference is thedimension of the problem. In matrix methods, stiffness andmass (as well as damping) are described via matrices in thecontext of finite elements with a dimension of several ofthousands. Thus it is possible to use these models for thewhole audible frequency range which is typically from 20Hz up to 20 kHz where a lot of non-pistonic andasymmetric modes occur (see Figure 5).

Another advantage of matrix methods is that we caninclude the influence of asymmetric effects due to loadingor due to interaction of the vibration system with theenvironment (think about attaching a speaker to anonsymmetrical cabinet).

The principle of using matrices for describing a physicaldomain is also used for the acoustical domain where theHelmholtz equation is solved by means of boundaryelements (see Figure 6).

As already discussed, a loudspeaker is a multiphysicsdevice, and the theory must account for that. Thus we haveto couple the mechanical systems discretized via finiteelements with the acoustical system discretized viaboundary elements (see Figure 7).

This leads to a very efficient simulation model to predictthe sound radiation of electromagnetic transducers.

Practical applications from industryIn this section we will present some examples from industryfor each different type of transducer as outlined previously.

WoofersThe sound pressure distribution of a typical wooferembedded in a cabinet is shown in Figure 8. Though awoofer covers the low frequency range it is also excited athigher frequencies. Thus, simulations are made up toseveral kHz (sometimes up to 20 kHz). If we comparesimulation and measurement we can see an excellentaccuracy of the proposed theory (see Figure 9).

TweetersThe basic principle of a tweeter is quite similar to that of awoofer, but a tweeter has to cover the high frequency range(simulations are made up to 40 kHz). Thus a tweeter ismuch smaller than a woofer. A typical sound pressuredistribution of a tweeter is given in Figure 10. Anotheradvantage of simulations can be seen in this example. Thesimulation also gives a view of the inner structure of thesystem. Measuring the sound pressure distribution insidethe housing of the tweeter would be nearly impossible.

HornsHorns are typically used for professional audio systems toreach a high sound pressure (based on a good efficiency)paired with a good frequency response and directivity (i.e.a constant sound radiation in all directions). An examplefor the sound pressure distribution inside a horn at twodifferent frequencies is given in Figure 11.

These models are kept very simple but they are veryefficient for calculating the directivity behavior.

Small TransducersDue to the tremendous business penetration of cell phonesthere has been a lot of development work in the last fewyears on small transducers. This has also been prompted

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Figure 8: Typical woofer embedded in a cabinet(with kind permission of CANTONElektronik GmbH & Co. KG)

“…a loudspeaker is a multiphysics device, andthe theory must account for that.”

Figure 7: Coupling of mechanical system and acoustical system

October 2005 9

by the transformation of a cell phone from a purespeech-based communication device into a truemultimedia device capable of playing music andvideo. Thus, special performance features arerequired for those mini loudspeakers.

Besides an excellent frequency response it is also veryimportant to radiate sound at reasonable levels, tolisten to music without headsets or for hands freeapplications. But on the other side there is a veryrestrictive constraint: the size. Cell phones dropped insize significantly in the 1990’s and so the speaker sizealso shrank. However, the easiest way to get areasonable radiated sound level is to increase size.This can be compensated by raising the excursion ofthe vibration system. But, as already outlined, themechanical system is nonlinear and this will lead tounwanted effects called distortion.

A typical CAD model of a small transducer for a cellphone can be seen in Figure 12. Note the size of 7x11mm. Figure 13 shows the mechanical nonlinearities. Ithas proved that matrix methods are very reliablemethods to significantly enhance the nonlinearbehavior of those small transducers within a verylimited time frame. Thus development time andsubsequently time to market can be reduced by a vastamount.

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Figure 9: Comparison of simulation (red) and measurement(blue), with kind permission of CANTON ElektronikGmbH & Co. KG

Figure 10: Sound pressure distribution on inner surfaces of a tweeter (with kind permission of CANTON ElektronikGmbH & Co. KG)

Figure 11:Sound pressure distribution inside a horn at two different frequencies (with permission of d&b audiotechnik AG)

“Measuring the soundpressure distributioninside the housing ofthe tweeter would benearly impossible…”

October 200510

Business BenefitsThe business benefits of matrix methods areobvious:

� deeper knowledge about the physics ofaudio devices

� significantly improved productperformance in terms of

�� audio performance

�� manufacturing costs

� reduced development time

� reduced time to market

� reduced development costs

However, these benefits can only be obtainedwhen carefully integrating this new technology intothe current development process of a company’sR&D department. Thus the implementation ofmatrix methods should also be done with respectto process flow and especially with respect totraining of the engineers involved. Buying softwareis, in that context, the simplest task. Transferringthe software to a highly efficient engineering toolis the real challenge.

Traps and PitfallsOne of the most sensitive steps in the successfulimplementation of matrix methods into thedevelopment cycle of a company is a carefullyplanned strategy for the introduction of the newmethod. From a technological point of view, mosttraps and pitfalls arise in the field of modeling.Modeling means to reduce the real application toa mathematical/physical model based on matrixmethods. A well balanced situation between beingtoo simple and being too complex must be found.Only then will the business benefits be realised.

Within that scope material properties are perhapsthe most important factors. Thus, fitting materialproperties of mathematical/physical materialmodels to real world behavior is the most serioustask. Engineers must have enough time to masterthis challenge, and this must be included in theoverall investment.

References[1] Panel Discussion: “FEA vs. Lumped Parameter

Modeling”, ALMA 2005 Winter Symposium,www.almainternational.org/wintersymposium2005

[2] AES Audio Engineering Society, www.aes.org

ContactAlfred J. SvobodnikNAD – Numerical Analysis & Design GmbH & Co as@NADwork.at

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“…matrix methods are very reliablemethods to significantly enhance thenonlinear behavior of those smalltransducers within a very limitedtime frame…”

Figure 12:Typical CAD model of a small transducer as used in cell phones

(with permission of Sonion Horsens A/S)

Figure 13:Mechanical nonlinear behavior of a small transducer (with permission of Sonion Horsens A/S)

solid line: displacement over force dashed line: displacement over stiffness