Flat loudspeaker enclosures

Daniel Beer1, Lutz Ehrig1, Lorenz Betz1, Marcel Schnemilich1, Waldemar Maysenholder21 Fraunhofer Institute for Digital Media Technology IDMT, 98693 Ilmenau, Germany,

Email: Daniel.Beer@idmt.fraunhofer.de2 Fraunhofer Institute for Building Physics IBP, 70569 Stuttgart, Germany,

IntroductionLoudspeaker enclosure design is usually considered inde-pendently of the geometry of the enclosure volume as longas standing waves are neglected. Based on a given setof Thiele-Small-Parameters only the size of the volumeis derived for a desired alignment. However, it hasbeen observed, that for enclosures having the same netvolume but different geometries, the vibration behaviorof an electrodynamic driver built in such enclosureschanged [1]. Especially by decreasing one dimension ofan enclosure, which results in a flat shape, the resonancefrequency of a driver built in that enclosure decreasedsignificantly.

Based on measurement results we would like to presentour findings and give explanations about possible causes.In future work the influence of the observed effects onthe acoustical behavior has to be investigated.

ObservationsTo identify the influence of the enclosure shape on thevibration behavior of an electrodynamic driver differentsetups were investigated. Each setup consisted of a closedbox, designed by alignment of acoustic suspension, andof one or four micro speakers (wired in parallel). Sincethe observed effects are more pronounced for the setupwith four micro speakers, the subsequent results will bepresented for this setup. In Table 1 the dimensions of

EnclosureDimensions [mm]

length width depth

1 122 120 422 290 210 103 466 268 5

Table 1: Dimensions of the first set of investigated enclosureswith a constant volume of 600 ml and four micro speakers

the investigated enclosures are specified, the net volume– 600 ml – is the same for each box. The loudspeakerboxes are depicted in Figure 1.

The vibration behavior of the electrodynamic driverswere investigated with the Klippel R&D-System. InTable 2 subsets of the linear parameter measurementsare given for each enclosure. Additionaly in Figure 2and Figure 3 the behavior of the electrical speakerimpedance and the membrane excursion are shown.According to the measurement results an influence ofthe enclosure shape can be recognized. The smallerthe enclosure depth the lower the resonance frequency.

Figure 1: First set of investigated enclosures with a volumeof 600 ml and four micro speakers wired in parallel

ParameterEnclosure depth

42 mm 10 mm 5 mm

fc [Hz] 281 253 220Qtc 0.62 0.63 0.73

Mmc [g] 0.59 0.69 1.06Kmc

[N/mm]1.84 1.73 2.03

Rmc [kg/s] 0.33 0.35 0.46

Table 2: Change of loudspeaker parameters (600 ml box,four micro speakers)

Between the 42 mm and the 5 mm depth enclosurethe difference is 61 Hz. Related to the membraneexcursion two regions can be localized. Above thebasic resonance frequency the excursion decreases withdecreasing enclosure depth. Below the resonance itbehaves vice versa. The parameter variation shown inTable 2 reveals that also the total q-factor Qtc, themoving mass Mmc, the stiffness Kmc and the mechanicallosses Rmc are affected. The moving mass is almostdoubled by reducing the depth from 42 mm to 5 mm.Due to standing waves some peaks occur in the curvesof the impedance and the excursion curve for the 10 mmand 5 mm enclosure.

Attempts of explanationPossible causes of the observations presented abovewill be discussed by the following three attempts ofexplanations, namely the classic loudspeaker theory, theviscous boundary layer effect and the inertia effect ofthe enclosed air.

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100 1,0000

20

40

60

Frequency [Hz]

Imp

edan

ce[O

hm

]

42 mm10 mm5 mm

Figure 2: Measured electrical impedance of the four parallelwired micro speakers in a 42 mm, 10 mm and 5 mm depthenclosure with a constant volume of 600 ml

10 100 1,0000

0.1

0.2

0.3

Frequency [Hz]

Mag

nit

ud

eof

TR

F[m

m/V

]

42 mm10 mm5 mm

Figure 3: Measured transfer function of membrane excursionof the micro speakers in a 42 mm, 10 mm and 5 mm depthenclosure with a constant volume of 600 ml

Attempt 1: Classic loudspeaker theory byThiele & SmallIn classic loudspeaker theory a simple loudspeaker con-sisting of a moving membrane and an enclosure, asdepicted in Figure 4, is simplified as a lumped elementmodel. Derived from that model the basic resonancefrequency fr is defined by

fr =1

2π·√

ss + svmm + 2mr

, [Hz] (1)

where ss is the stiffness of suspension, sv is the stiffnessof the air volume inside the enclosure, mm is the totalmass of diaphragm, voice coil and carrier and mr is themass of the air load [2].

The stiffness of the air sv is given by

sv =ρ0 · c20 ·A2

Vb. [N/m] (2)

enclosure

suspension (ss) diaphragm(A, mm, r) air volume

(Vb, sv, ρ0, c0, mr)

Figure 4: Schematic loudspeaker model with importantparameters

Only the size of the volume is considered by Vb, not theshape of that volume. ρ0 and c0 are the density of airand speed of sound in air respectively, A is the size of thediaphragm. The mass of the air load mr is defined by

mr|kr≤1 = 1.93 · r3 · ρ0 , [kg] (3)

with the diaphragm radius r and the wave number k [2].

From the equations above it can be seen, that neither thedefinition of stiffness nor the definition of mass considersthe shape of the gas volume inside the enclosure. Thuswith classical loudspeaker theory the observed effectscannot be explained.

Attempt 2: Viscous boundary layer effectsBetween the gas molecules with the velocity v = v0and an enclosure wall with a velocity of v = 0 frictionoccurs. Because of this friction a smooth change invelocity occurs close to the wall. This causes a velocitydistribution as shown in Figure 5. The molecule velocity

enclosure walls molecule velocity (v(y))

boundary layer (λv)

Figure 5: Velocity profile of gas molecules in a flat enclosure

slows down to zero for decreasing distances to the wall.This region is called boundary layer (λv). The size ofthis layer can be calculated by

λv = 2π

√2η

ωρ0≈ 1.4√

f, [cm] (4)

where η is the dynamic viscosity of air and ω is theangular frequency [3]. For a frequency of 100 Hz theboundary layer is about 1.4 mm. Regarding very flatenclosures, e. g. 5 mm depth, does this boundary layeraffect the vibration behavior of the diaphragm?

Numeric simulations have shown that the resonance fre-quency of a harmonic oscillator decreases by considering

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boundary layer effects in the governing equations. How-ever, the influence is not as high as in the measurementresults.

Attempt 3: Inertia effect of the enclosedairBasic idea: At low frequencies the enclosed air reactsto the motion of the diaphragm like a spring. The air-spring stiffness depends on the volume of the air spacebut not on its shape. Usually this spring is treatedas ”ideal”, i. e. its mass is neglected. If the mass ofthe diaphragm is small, this may no longer be justified.In a one-dimensional calculation for a cylindrical tubewith the diaphragm as a piston on one end and a rigidtermination at the other end the kinetic energy of theenclosed air is easily obtained. It can be accounted foras an effective air mass which is equal to one third ofthe total air mass. In equation (1) for the resonancefrequency this effective air mass would replace one of thetwo air loads mr.

If the shape of the air space is different, the one-dimensional model is no longer applicable. However, asa rough approximation, one can conceive the air spaceas containing two one-dimensional springs, one actingnormal to the diaphragm (”normal spring”), the otherone acting perpendicular to it (”lateral spring”). Thismodel still assumes – corresponding to an infinite speedof sound – that the sound pressure in the air space iseverywhere the same at a certain time. Therefore, wavepropagation effects are not included.

Motion of the diaphragm displaces a time-varyingamount of air, which is split into two parts, one for eachspring. In the case of a very flat enclosure with the samevolume the normal spring is much stiffer than the lateralspring, since the spring stiffness is inversely proportionalto the spring length. Consequently most of the displacedair moves into the softer lateral spring. Compared to thecylindrical case this implies larger air motion and hencea larger effective air mass, which qualitatively explainsthe shift to lower resonance frequencies.

Results of simulations and measurements: Thisinterpretation of the measured findings as an inertiaeffect is confirmed by finite element calculations whichyield corresponding results without accounting for vis-cous boundary layer effects. As mentioned above it wasfound, that this effect appears not to be that prominent,at least as long as the enclosure depth is not smaller than5 mm.

For numerical calculations the acoustics module in Com-sol Multiphysics was used. A simple model whichconsists of an oscillating plate coupled to a closed volumewas built. The plate is supported by a spring foundationrepresenting the stiffnes of the diaphragm suspension ofthe loudspeaker driver and was excited by a sinusoidalforce with constant amplitude over frequency.

Investigating the inertia effect of the enclosed air anotherset of enclosures, as specified in Table 3, was simulated.For the enclosures 5 and 6 the depth is kept constant

EnclosureDimensions [mm]

length width depth

4 60 60 425 203 150 56 600 50 5

Table 3: Dimensions of the second set of investigatedenclosures with a constant volume of 150 ml

and the shape of the enclosure is varied especially inone dimension. As predicted above, due to the highlength of enclosure 6 and therefore a very soft ”lateralspring” the calculated resonance frequency drops from268 Hz (enclosure 5) to 219 Hz as shown in Figure 6.The resonance frequency of enclosure 4 is 282 Hz.

100 200 300 4000

0.2

0.4

0.6

0.8

1

Frequency [Hz]

Nor

mal

ized

dis

pla

cem

ent

[1]

42 mm5 mm

5 mm (long)

Figure 6: FEM simulation of normalized displacement overfrequency (enclosure 4 (42 mm), 5 (5 mm) and 6 (5 mm, long),150 ml)

In Figure 7 and Figure 8 the velocity is shown as coloredslices and vector fields for enclosure 4 and enclosure 6.It can be seen that in enclosure 4 there is no preferedorientation of the movement of the air. In contrast,the movement of the air in enclosure 6 has a preferedorientation in the direction of the largest dimension. Itexceeds the largest dimension of enclosure 4 many timesover. Thus the moved mass of air in enclosure 6 ishigher compared to that in enclosure 4. Consequently the

Figure 7: FEM model of enclosure 4 (42 mm), shown is thevelocity as colored slices and vector field

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Figure 8: FEM model of enclosure 6 (5 mm, long), shown isthe velocity as colored slices and vector field

Figure 9: Second set of investigated enclosures with a fixedvolume of 150 ml and one micro speaker

resonance frequency of enclosure 6 is significantly lowerthan for enclosure 4 as shown in Figure 6.

The simulated second set of enclosures was built asshown in Figure 9 and measurements were performed.According to the simulations the moved mass of air Mmc

increases especially for enclosure 6 (Table 4). Conse-quently the resonance frequency decreases as shown inFigure 10.

Summary and outlookIt has been shown that the shape of a loudspeakerenclosure cannot always be neglected and may havea noticeable influence on the vibration behavior of aloudspeaker driver. Especially huge differences for at

ParameterEnclosure depth

42 mm 5 mm 5 mm(long)

fc [Hz] 268 242 195Qtc 0.64 0.61 0.69

Mmc [g] 0.18 0.19 0.33Kmc

[N/mm]0.50 0.43 0.49

Rmc [kg/s] 0.07 0.12 0.16

Table 4: Change of loudspeaker parameters (enclosure 4(42 mm), 5 (5 mm) and 6 (5 mm, long), 150 ml)

100 1,0000

100

200

300

400

Frequency [Hz]

Imp

edan

ce[O

hm

]

42 mm5 mm

5 mm (long)

Figure 10: Measured electrical impedance of the driver inenclosure 4 (42 mm), 5 (5 mm) and 6 (5 mm, long), 150 ml

least two dimensions of an enclosure, e. g. very thin andvery long, lead to an increased moving mass and thereforeto a decreased resonance frequency.

In future work the question whether any benefit canbe drawn from this and whether the lower resonancefrequency leads to a lower cut-off frequency will beinvestigated. If lightweight diaphragms are advantageousconcerning the chassis design and if the acoustic behavioris affected at all has to be analyzed as well.

References[1] Beer, D.: Untersuchungen zu Flachlautsprechern an

schallreflektierenden Grenzflachen; Dissertation, TU-Ilmenau, 2011

[2] Sahm, H.: Arbeitsbuch fur Lautsprecher-Systeme;Franzis-Verlag GmbH, Munchen, 1987

[3] Meyer, E.; Neumann, E.-G.: Physikalische undTechnische Akustik; Vieweg & Sohn Verlag GmbH,Braunschweig 1975

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