Measurement of the four pole matrix of a sample in a transmission tube
Rene BOONEN(1)(1) Nabla Technical Consulting, Germany, Boonen.Rene@gmail.com

Abstract
A transmission tube has been developed to measure the four pole matrix of a sample of absorbing material.
The electrical analogy of a sample in a transmission tube can be considered as a T-equivalent circuit of three
impedances, two in series and one in parallel. As consequence, three measurements with three different closing
impedances needs to be carried out. From these measurements, a set of equations will result from which the three
impedances of the T-equivalent circuit can be determined in real and imaginary parts in terms of frequency. Oncethese impedances are known, any four pole matrix, such as the transfer or the scattering matrix, of the sample canbe set up. The method has been validated by measurements of several sample combinations in the impedance
tube. A good agreement has been established between transmission tube and impedance tube measurements ina range of 40Hz-4kHz.
Keywords: Kundt, Transmission, Impedance, Transfer, Matrix

1INTRODUCTION

The Kundt’s tube is used to measure the acoustical absorption, reflection or impedance of a sample of absorbing
material or a muffler. The sample or muffler needs to be backed by a known impedance, such as a hardwall, an open end or an anechoic end. However, if one wants to calculate layers of absorbing materials or
muffler systems, four pole data such as the transfer matrix are needed. Therefore, a transmission tube has
been developed which measures the four pole data with high accuracy. The sample is represented as a Tequivalent circuit [1] of three impedances, two in series and one in parallel. Three impedance measurementswith three different closing impedances will be carried out according to the ISO 10534-2 standard [2]. Fromthese measurements, a set of equations will result from which the three impedances of the T-equivalent circuit
can be determined in real and imaginary parts in terms of frequency. Once these impedances are known, any
four pole matrix, such as the transfer or the scattering matrix, of the sample can be set up. The method has
been validated by measurements of several sample combinations in the impedance tube.

2SETUP OF THE TRANSMISSION TUBE

Figure 1 presents a scheme of a normal Kundt tube using the two microphone transfer function method according to the ISO 10534-2 standard.

Figure 1. Scheme of an impedance tube using the two microphone transfer function method according to theISO 10534-2 standard.

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The wave pattern in the wave guide is governed by the one-dimensional Helmholtz wave equation, which describes the pressure distribution along the wave guide [3]. At each position x, the pressure p in terms of theangle frequency ω in the wave guide is expressed as:p(x, ω) = φg

Z0 Zge− j k l(e j k x + Γl e− j k x )Z0 + Zg 1 − Γl Γg e− j 2 k l(1)wherein φg the flow and Zg the internal impedance of the acoustic source, Z0 the characteristic tube impedance,
k the wave number, l the length of the tube, Γl and Γg the reflection coefficient on the load and the source
impedance respectively.
By taking the transfer function T12 between the acoustic pressures at two positions x1 and x2 , all the quantities
mentioned above disappear except the reflection coefficient at the load. Consequently, this reflection coefficientΓl can be determined. The load impedance Zl is then determined from the reflection coefficient Γl .
T12 =p(x1 , ω) e j k x1 + Γl e− j k x1
=p(x2 , ω) e j k x2 + Γl e− j k x2

e j k x1 − T12 e j k x2
Γl = − − j k x
1 − T12 e− j k x2e⇒

Zl = Z0(2)1 + Γl1 − Γl(3)The load impedance Zl is the impedance of everything appearing behind the reference section. By taking thetransfer function T12 , the source reflection coefficient is eliminated. This implicates that the structures in thesource tube, i.e. everything between the microphone at position x2 and the sound source is eliminated from theexpression of the load impedance. These properties form the basis of the newly developed transmission tube.

Figure 2. Scheme of the transmission tube with two microphone heads, sample ZS and closing impedance ZA .

A scheme of the transmission tube is presented in figure 2. In this tube, a second microphone arrangement
has been introduced between the sample ZS and the closing impedance ZA . The measurement head with microphones 1 and 2 measures the impedance behind the first reference section, consisting of the sample impedance,
the impedance of the second measurement head and the closing impedance. The measurement head with microphones 3 and 4 measures the impedance behind the second reference section, consisting of only the closing
impedance ZA .
To determine the three impedances Z1 , Z2 and Z3 , three impedance measurements at each measurement head
have to be carried out with three different closing impedances using the ISO 10534-2 two microphone transferfunction method. From these measured impedances, a set of three algebraic equations will be set up from which
Z1 , Z2 and Z3 will be solved in amplitude and phase. Once these three impedances are known, any four polematrix such as the impedance matrix, the admittance matrix, the transfer matrix and the scattering matrix canbe obtained.

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The transfer matrix H =ACBD
can be determined from the three impedances Z1 , Z2 , Z3 and the character-istic impedance Z0 :Z1Z0 Z1 Z21+Z0Z1+Z0Z2+
Z3Z3H=
Z211+Z3 Z0Z3The transmission loss T L can then be calculated from transfer matrix:
BT L = 20 log(.5 (A ++C Z0 + D))
Z0


3(4)

(5)

EXPERIMENTAL RESULTS USING THE TRANSMISSION TUBE

Figure 3 presents the newly developed transmission tube.

Figure 3. Transmission tube with internal diameter of 45 mm with two measurement heads and closing
impedance. Total length is 1.4 m

At the left end situates the loudspeaker, then the first measurement head with two microphones, then the sample
holder, then the second measurement head with the closing impedance tube at the right side. The closing
impedance tube has a valve system to create the three closing impedances without disassembling the setup. The
transmission tube has a frequency range from 40 Hz until 4 kHz.
The test sample is presented in figure 4. It is a glass fiber sample of 20 mm thickness which fit slightly tightin the sample holder with a surplus of 0.2 mm.
The figures 5, 6 and 7 presents the measurement results of the three impedances Z1 , Z2 and Z3 of the Tequivalent circuit between 40 Hz and 4 kHz. The two series impedances Z1 and Z2 are more or less resistiveof nature. Their magnitude is mainly horizontal. The phase remains around zero which indicates the resistivenature. At 800 Hz, a sample resonance occur and above this resonance, the magnitude is slightly increasing andthe phase tends towards +90 o , indicating that the sample behaves more as an acoustic inertia. The impedanceZ3 behaves as an acoustic volume. The magnitude decreases with a slope of −20 dB/decade and with a phaseof −90 o .
From these three impedances, the transfer matrix H is calculated using expression (4). Then, the transmission
loss of this sample is determined using expression (5) and is presented in figure 8. Once the transfer matrix
has been obtained for a sample, they can be collected in a library of sample transfer matrices. These transfermatrices can be used to predict the acoustic reflection and absorption of different assemblies of absorption layers
without prior measurements with the impedance tube.

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40180209000-20-90-40102

103Phase (deg)Amplitude Z1 (dB)Figure 4. Sample of glass fiber material with diameter 45mm and 20mm thickness.

-180

frequency (Hz)Figure 5. Measured impedance Z1 from the T-equivalent circuit. (magnitude in red line with reference 0 dB isZ0 , phase in blue line).

To illustrate this possibility, as a first case, the transfer matrix T of the sample against a hard wall will beobtained by multiplying the measured transfer matrix by the transfer matrix of a hard wall:1 0A BT=(6)11C DZHwherein ZH is the hard wall impedance, which is set to 1000 times the characteristic impedance Z0 .
The impedance ZT of the sample against the hard wall is then determined from the the elements T(1, 1) andT(2, 1) of the transfer matrix T:
T(1, 1)
ZT =(7)
T(2, 1)
The reflection and absorption coefficient Γ and α are then determined as:
ZTZ −1

Γ = Z0T
Z04972+1and

α = 1 − Γ Γc ;

(8)

180

2090

00-20-90-40
102103Phase (deg)Amplitude Z2 (dB)

40-180frequency (Hz)6018040902000-90-20102103Phase (deg)Amplitude Z3 (dB)Figure 6. Measured impedance Z2 from the T-equivalent circuit. (magnitude in red line with reference 0 dB is
Z0 , phase in blue line).

-180

frequency (Hz)Figure 7. Measured impedance Z3 from the T-equivalent circuit. (magnitude in red line with reference 0 dB is
Z0 , phase in blue line).

To evaluate the result, a benchmark measurement has been carried out in a classical impedance tube. In thesample holder, the sample has been put directly against the hard wall. Then, the reflection and absorption
coefficient are measured and plotted against the predicted coefficients by using the transfer matrix. Figure 9
presents the result. In red, the real part of the reflection coefficient obtained from the transfer matrix is plottedagainst the classical tube measurement in green. In dark blue, the imaginary part of the reflection coefficientobtained from the transfer matrix is plotted against the classical tube measurement in light blue. Both plotsshow a remarkable agreement.
Figure 10 presents the absorption coefficient. In red, the absorption coefficient obtained from the transfer matrix
is plotted against the classical tube measurement in green. Also these plots show a remarkable agreement.
As a second case, an assembly consisting of the sample and an air gap of 100 mm closed by a hard wall will
be considered. The total transfer matrix T of the assembly will be obtained by multiplying the measured sample

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20TransmissionLoss (dB)151050-5-1010210

3frequency (Hz)Figure 8. Transmission loss of the sample presented in figure 4 between 40 Hz and 4 kHz.

1reflection

0.50-0.5-1102103frequency (Hz)Figure 9. Comparison between the measured reflection coefficient of the sample in the impedance tube andthe predicted reflection coefficient from the transfer matrix of the sample measured in the transmission tube inreal/imaginary parts.

transfer matrix by the transfer matrix of a wave guide of 100 mm air gap en then by the transfer matrix of a
hard wall:
cos k Lj Z0 sin k L
1 0
A B
T=(9)
j11cos k L
C DZHZ sin k L
0Then, the reflection coefficient and the absorption coefficient will be determined again by using the expressions (7) and (8). The same assembly will be realized in the sample holder of the impedance tube. Between
the sample and the hard wall situates an air gap of 100 mm. Figure 11 presents the resulting reflection coefficient. In red, the real part of the reflection coefficient obtained from the transfer matrix is plotted against theclassical tube measurement in green. In dark blue, the imaginary part of the reflection coefficient obtained fromthe transfer matrix is plotted against the classical tube measurement in light blue. Again, both plots show aremarkable agreement.

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1absorption

0.8

0.6

0.4

0.2

0102103

frequency (Hz)Figure 10. Comparison between the measured absorption coefficient of the sample in the impedance tube andthe predicted absorption coefficient from the transfer matrix of the sample measured in the transmission tube.

1reflection

0.5

0-0.5-1
102

103

frequency (Hz)Figure 11. Comparison between the measured reflection coefficient of the sample in the impedance tube andthe predicted reflection coefficient from the transfer matrix of the sample measured in the transmission tube inreal/imaginary parts.

Figure 12 presents the absorption coefficient. In red, the absorption coefficient obtained from the transfer matrix
is plotted against the classical tube measurement in green. Also these plots show a remarkable agreement.
These benchmarks prove that is possible to reconstruct reflection and absorption coefficients in several assemblies by manipulating transfer matrices of samples measured in this newly developed transmission tube.

4CONCLUSIONS

A new method for determination four pole parameters by impedance measurements has been presented. Thesample is represented by a T-equivalent circuit of three impedances. These three impedances will be identified
from three impedance measurements with three different closing impedances. The measurements are conducted4975

1absorption

0.8

0.6

0.4

0.2

010

210

3

frequency (Hz)Figure 12. Comparison between the measured absorption coefficient of the sample in the impedance tube andthe predicted absorption coefficient from the transfer matrix of the sample measured in the transmission tube.

in such a way that the source impedance does not appear in the measurements. No prior calibrations of closing
impedances are required. It has become possible to collect material transfer matrices in a library, from whichreflection and absorption coefficients for several assemblies can be predicted.

ACKNOWLEDGEMENTS
The author whishes to thank Mark Brennan and Jan Vandenbroeck from Huntsman Polyurethanes for evaluating
the transmission tube.

REFERENCES[1] Terman, F.E., Electronic and Radio Engineering, McGraw-Hill Kogakusha Ltd., 1955.
[2] ISO, International Standard ISO 10534-2: Determination of sound absorption coefficient and impedance in
impedance tubes – Part 2: Transfer-function method, 1998.
[3] R. Boonen; P. Sas; W. Desmet; W. Lauriks; G.Vermeir; Calibration of the two microphone transfer function
method by determining the hard wall impedance at shifted reference sections. Proceedings of the International Conference on Noise and Vibration Engineering, Leuven, Belgium, 2008, Vol. 3; pp. 1413 - 1424

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