PROCEEDINGS of the23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Measurement of the Back Loading of a Loudspeaker Mounted on aClosed Enclosure - A First Approach1

Jorge N. Moreno; 2Richard Rivera; 3Luis Ángelo Velarde

1,2

Acoustic Laboratory, Physics Departament, Pontificia Universidad Católica del Perú
3Engineering Departament, Pontificia Universidad Católica del Perú

ABSTRACTIn this paper, a simple way to measure the parameters related to the back loading of a loudspeaker
mounted on a closed enclosure fully fill with filling material is presented. If the electro -mechanoacoustical parameters of the loudspeaker in free air are known or can be measured, then by measuring
one more time the parameters of the loudspeaker but this time mounted on a closed enclosure, it ispossible to calculate the back-loading parameters due to the enclosure, including the filling material.
This method could also be used to measure the effect of filling material in the back cavity of midrange
units, tweeters and compression drivers.
Keywords: Enclosure, acoustic impedance filling material,
1.

INTRODUCTION

Determine the acoustic impedance of the filling material of a loudspeaker enclosure always has
been a tedious task (1). After adding the filling material and the mechanical reinforcements to stiffen
the enclosure, it turns that the calculation of the box volume becomes uncertain. It was shown in (2)that the mechanical impedance of the loudspeaker including the air load reflected to the electrical sideis equal to velocity to current ratio multiplied by the force factor. In this paper such ratio will bemeasured in order to work without the influence of the electrical part of the loudspeaker. A laser
method will be used to determine the parameters of the loudspeaker. In order to perform thecalculations, the parameters will be measure first in free air and then mounted on closed enclosure
without filling material.

2. ISOLATING THE MECHANICAL IMPEDANCEAs it was mention before it is possible to isolate the mechanical mobility of the loudspeaker without
the influence of the electrical part, Figure 1 shows an example of such function. Figure 2 shows
respectively the real and imaginary parts of the mechanical impedance, both results shown in thefigures correspond to a JBL 2206 12” loudspeaker, this function shows at low frequencies theviscoelastic effect of the suspension.

vBli7310

Z ES(1)

Figure 1 - Mechanical impedance reflectedto the electrical side of a JBL 2206.

Figure 2 - Real and Imaginary part of themechanical impedance of a JBL 2206.

3. THE MODELS
Figure 3 and 4 show respectively the well-known impedance type equivalent circuits of aloudspeaker including the viscoelastic behavior of the suspension in free air and mounted on a closed
enclosure, not considering the electrical side. The equivalent circuit shown in Figure 4 include s asimplified model of the filling material valid at low frequencies and the compliance of the air in theenclosure full of filling material. The complete model of the filling material can be found in (3) and(4). Since the circuit in Figure 4 can be simplified to a similar circuit as the one shown in Figure 3,
the laser method can be used to measure the parameters of the loudspeaker mounted first on an empty
enclosure and then in one with filling material. Since the parameters of the loudspeaker in free air has
previously been measured, then the parameters of the closed enclosure can be calculated. The
equivalent circuits shown in figure 3 and 4 are suitable to describe the measurement pre sented in thispaper.
RMS1

C MS1

RMS 2

C MS 2

Figure 3 – Mechanical equivalentcircuit of a loudspeaker in free air
including viscoelasticity.

Figure 4 – Mechanical equivalent circuit of aloudspeaker including viscoelasticity mounted ona fully filled enclosure.

Were:
F(t)

Force driving the mechanical part NMMD

Mass of the driver not including the air load kg.

RMS1

Mechanical resistance component (TGM) Voigt branch N.s/m

CMS1

Mechanical compliance component (TGM) Voigt branch m/N

RMS2

Mechanical resistance component (TGM) Maxwell branch N.s/m

CMS2

Mechanical compliance component (TGM) Maxwell branch m/N

MMAF

Mass load of the air in the fiber-filled area on the transducer Kg

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MMF

Mass of the fibers kg

RMfl

Mechanical resistance due to flow resistance N.s/m

CMAF

Compliance of the air in the fiber filled area m/N

4. RELATIONSHIP BETWEEN THE MECHANICAL RESISTANCE AND THERESONANCE FREQUENCY IN DIFFERENT CONDITIONSWhen the parameters of the loudspeaker in free air are measured, the mechanical resistance iscalculated at the resonance frequency. The traditional model of a loudspeaker assumes that themechanical resistance is constant within the piston range of the loudspeaker, but this is not true. Figure
5 shows the measured mechanical resistance of a JBL 2206 12” loudspeaker superimposed with theconstant resistance. Let us assume that the resonance frequency of the loudspeaker in free air is fs.
When the loudspeaker is measured in a closed enclosure without filling material, the resonant
frequency shift to the right toward the frequency fc due to the change in compliance, it is easy tonotice that the mechanical resistance diminishes. When the loudspeaker is measured in an enclosure
with filling material, the resonance frequency shift again but this time to the left towards fcf because
the loudspeaker now “sees” a larger volume and now the mechanical resistance increases. Figure 6
shows the mobility of this loudspeaker for the 3 cases mention above.

211 3 2

Figure 5 - Mechanical impedance of a2206 JBL 12”loudspeaker

3Figure 6 - Velocity to current ratio of a JBL 2206

1. Resonant frequency of the loudspeaker in free air (fs)
2. Resonant frequency of the loudspeaker mounted on a closed enclosure without filling material(fc)
3. Resonant frequency of the loudspeaker mounted on a closed enclosure fully fill ed with fillingmaterial (fcf)

5. MEASUREMENT OF THE PARAMETERS OF THE LOUDSPEAKERS IN FREE
AIR AND MOUNTED ON A CLOSED ENCLOSURE WITH AND WITHOUT FILLING
MATERIAL
5.1 Volume calculation
The laser method is used to measure the parameters of the loudspeakers. Three measurements have tobe performed in order to do the calculations: In free air, mounted on an empty enclosure and in one
fully filled with filling material. Using the first two measurement, the volume of the box without
filling material can be calculated, using the next two measurements, it is possible to calculate thecharacteristics of the filling material.

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Table 1

Table 2

Free Air Parameters

Empty Parameters

12 "6.5" Full

8" Full

12 "6.5" Full

8" FullWoofer

Range

Range

WooferRangeRange

Sd(m2)

0.04750.01340.0219Sd(m2)

0.04750.01340.0219Re(Ohm)

5.90

7.50

6.2

Re(Ohm)

5.90

7.50

6.2

fs(Hz)

50.26

94.26

59.303fc(Hz)

62.04

136.67123.92

Bl (Txm)

15.89

5.28

8.883

Bl (Txm)

16.23

5.15

8.62

Qms

2.90

3.80

1.87

Qmc

3.47

6.13

3.9

Qes

0.46

1.00

0.427

Qec

0.57

1.55

0.913

Qts

0.39

0.79

0.347

Qtc

0.49

1.24

0.737

Mms (g)

61.55

6.33

14.63

Mmc (g)

64.70

6.43

14.13

Rms (Kg/s)

6.73

1.00

2.96

Rmc (Kg/s)

7.26

0.907

2.83

Cms (um/N)

163455494

Cmc (um/N)

102

211

116.67

Vas (l)

51.90

11.37

33.767

Vac (l)

32.00

5.27

7.967

No (%)

1.41

0.92

1.603

No (%)

1.29

0.843

1.617

Table 3

6.5"

Filled Parameters
12 "6.5" Full

8" FullWoofer

Range

Range

Calculated

Sd(m2)

0.04750.01340.0219from measurement (L)Re(Ohm)

5.80

7.50

6.2

filled box

fcf(Hz)59.92

131.94

119.43Calculated

Bl (Txm)

16.22

5.39

9.02

from measurement (L)

Qmcf

2.81

4.18

2.04

empty box

Qecf

0.56

1.40

0.82

Calculated

Qtcf

0.47

1.05

0.64

using a ruler (L) empty

Mmcf (g)

67.90

6.60

14.3

Error (%) for empty

Rmcf (Kg/s)

9.11

1.30

3.79

box case

Cmcf (um/N)

104

222

124

Vacf (l)

33.40

5.50

1.71

No (%)

1.25

12 "Full8" FullWoofer

Range

Range

88.1

10.76

10.82

86.40

10.05

10.31

86.45

10.5

10.5

0.057

4.29

1.81

volumevolumevolumeUsing the first and the third measurement the increase of volume “seen” by the loudspeaker due tofilling material can be calculated. Tables 1, 2 and 3 show the results of the measured parameters andTable 4 shows the results of the calculated volumes in empty box and filled box. The volumes found
for the empty box case, are compared with the volumes measured with a ruler.
The volumes of the boxes without filling material were calculated using the traditional parameters7313

and the traditional equivalent circuit, considering C ms and C mc in series. After removing thecompliance of the loudspeaker, the acoustical compliance of the box was calculated, then the volumes
are calculated using the well known definition of compliance (5):CAB

V / Uc2(2)Were:ρ = 1.18 m/s
c = 340 m/s
The volume seen by the loudspeaker in the box fully filled with filling material is calculated withthe same procedure.
5.2 Filling material parameters
By measuring the velocity to current ratio and then the reciprocal value of this parameter multiplied
by Bl, it is possible to find the mechanical impedance first and then the acoustical impedance for thethree cases previously described. Figure 7 shows the resistance and the imaginary part of theacoustical impedance of the 8” loudspeaker described in tables 1, 2 and 3 in free air, green function,
in a closed enclosure without filling material, red function and in an enclosure fully filled with fillingmaterial, orange function.
The boxes were carefully built so the resistance of the loudspeaker in free air is a lmost
superimposed to the resistance of the box without filling material, which means that the box introduces
almost no losses.

Figure 7 - Mechanical impedance of and 6.5”
speaker in free air, and in a box without andwith filling material (green, red and orangerespectively).

Figure 8 - Acoustic impedance between points A
and B shown in Figure 4, superimposed results
obtained using a 6.5” and 8” speaker (green andred curves respectively).

Figure 8, is the subtraction of the acoustic impedance of the fully filled case minus the acoustic
impedance of the empty enclosure case; these curves were calculated with measurements done using
the same 10.5-liter enclosure with the same amount of filling material. The acoustic impedances usedfor these calculations were measured respectively with the 6.5” and 8” loudspeakers previously
described, green and red curves. Almost the same values were obtained with the two loudspeakers.
These curves represent the impedance between the points A-B in the circuit shown in Figure 4.

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Figure 9 – JBL 2206 case.

Figure 10 – JBL2206 case.

The curves showed in figures 9 and 10 correspond to a 12” JBL 2206 not described in the tables
above. The green curves correspond to the loudspeaker mounted on an 86.45 -liter enclosure fully
filled with filling material; the red functions correspond to the acoustic impedance between points AB in figure 4.
Figure 9, shows the magnitude and phase for both cases; it can clearly be observed how themagnitude increases for the case were the loudspeaker is mounted on the filled enclosure d ue to thecompliance of the box green curve and how its phase is -90 degrees with respect to the filling material
alone red function.
Figure 10, shows the real and imaginary parts for the previous case, it can clearly be observe d thatthe real part of the box coincide with real part of the filling, so the increase in the magnitude in figure
9 is only due to the compliance of the box while the resistive part remain the same.

6. LOUDSPEAKER MODELING USING THE FILLING MATERIAL CURVE
With the information presented in tables 1, 2 and 3 it is possible to calculate a single value torepresent the mechanical or acoustical resistance of the filling material and then replace this value inthe loudspeaker model, or, use the magnitude and phase response of the filling material to model theloudspeaker. Figure 11 show the result of modeling the sound pressure level, the magnitude of theelectrical impedance and the voice coil displacement of the 8” loudspeaker in a 10.5-liter box. Redcurves represent the results obtained with a single value 1278 acoustic Ohm, the green curves werecalculated with the magnitude and phase response of the acoustic impedance shown in figure 8. It isclear that the results are different.

Figure 11 – Loudspeaker simulation

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7. CONCLUSIONSIt has been shown that by measuring the velocity to current ratio of a loudspeaker in free air, in anempty enclosure and in a fully filled enclosure, it is possible to determine the volume of the empty
enclosure, the volume of the box with filling material and the characteristics of the filling material.
Further research has to be done to be able to measure the characteristics of partially filled
enclosures.

ACKNOWLEDGEMENTS
Authors want to thank Celso Llimpe for his help in the preparation of this paper, to Ayrton Urviola
for his help in the calculations, to Paul Castro for his help in the measurements in the impedance tube,and to Victor Centa for the loudspeaker modeling.

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