Determination of transmission and insertion loss for the general multi-inlet multi-outlet case.
Prior research on assessing multiple inlet and outlet muffers is limited, and only recently have researchers begun to consider suitable metrics for multiple inlet and outlet muffers. In this paper, transmission loss and insertion loss are defined for multiple inlet and outlet muffers using a superposition method that can be extended to any m-inlet n-outlet muffer. Transmission loss is determined assuming that the sources and terminations are anechoic. On the other hand, insertion loss considers reflections. For both metrics, the amplitude and phase relationship between the sources should be known a priori. This paper explains both metrics, and measurement of transmission and insertion loss are demonstrated for a 2-inlet 2-outlet muffer with good agreement.
CITATION: Zhang, Y., Herrin, D., Wu, T., and Hua, X., "Determination of Transmission and Insertion Loss for the General Multi-Inlet Multi-Outlet Case," SAE Int. J. Passeng. Cars - Mech. Syst. 9(1):2016, doi:10.4271/2016-01-1310.
Most prior muffer research has been dedicated to the single-inlet and single-outlet (SISO) case. Transmission and insertion loss are the two metrics most commonly used to evaluate the performance of a muffer system. For a SISO muffer, transmission loss is defined as the logarithmic ratio between incident and transmitted sound powers, under the assumption that both the source and termination are anechoic. This eliminates the influence of reflections from the source and termination, thus making transmission loss an attribute of the muffer itself . Transmission loss can be readily calculated from the transfer matrix below the plane wave cut-off frequency.
Insertion loss, on the other hand, is defined as the difference between the sound pressure levels with and without the muffer installed. Insertion loss is an evaluation of the muffer in a specific source and termination combination and is most easily measured. However, calculation is problematic. Refections from the source and termination must be taken into consideration meaning that the source and termination impedances must be known to calculate insertion loss .
Multi-inlet and multi-outlet (MIMO) muffers have been investigated in generally two ways. Selamet and Ji  and Denia et al.  investigated the transmission loss of circular expansion chambers using a mode-matching approach and developed analytical solutions for pre-defined configurations. The solutions from a mode-matching approach are for certain configurations and cannot be easily extended to the general case. Another approach by Jiang  and Mimani and Munjal  is based on an impedance matrix, which is obtained by either plane wave analysis or the boundary or finite element method. In both approaches, it is noted that transmission loss and insertion loss for MIMO muffers are dependent on the amplitude and phase relationship between the sources, which can be dealt with by using complex ratios between each source and a reference source.
In this work, the definitions for transmission loss and insertion loss are extended to the MIMO case using a different approach based on transfer matrix theory and superposition. A MIMO muffer is considered as several SISO muffers, and transfer matrix theory is used to characterize each. Results are combined using a superposition approach assuming that the phasing between sources is known. The approach is validated using an experimental 2-inlet 2-outlet muffer.
For MIMO muffers, the amplitude and phase relationship between sources and the source impedance must be taken into consideration when defining transmission and insertion loss. In this work, a circuit analogy model  is used to describe the sources. The sound source is modeled as a pressure source (analogous to a voltage source) and source impedance ([z.sub.S]) in series with the acoustic load impedance ([z.sub.L]) (Fig.1). It is assumed that the particle velocity (analogous to electrical current) is continuous at the source-load interface.
From this model, it can be observed that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where [p.sub.S] and [z.sub.S] are the source strength and impedance respectively. [p.sub.L] and [z.sub.L] are the respective load sound pressure and impedance. To use this model, the interface between the sound source and load must be assumed. [p.sub.L] and [z.sub.L] can be determined indirectly from measurement by performing wave decomposition downstream of the source using the two-microphone method .
As there are two unknowns ([p.sub.S] and [z.sub.S]) in Eqn.1, at least two equations should be obtained from two different working conditions of the system (i.e., acoustic loads). The set of equations can be represented as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
where the superscript indicate the respective load. For higher accuracy, more than two loads can be used and the source properties can be solved using a least-square method .
The definition of transmission loss for MIMO muffers is a straight forward extension from the SISO case. As Fig. 2 shows, the sound pressures inside Inlet j are decomposed into incident and reflected waves [p.sub.i,j] and [p.sub.r,j] respectively. Under the assumption that all the sources and terminations are anechoic, the transmission loss is defined as the ratio between the summation of incident sound power in the inlets and the summation of transmitted sound power in the outlets where [s.sub.j] and [s.sub.k] are cross-sectional area of inlets and outlets. Accordingly,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
To calculate [p.sub.i,j] and [p.sub.k], a superposition model can be used. First assume only one source (j) is active (Fig. 3), the transmitted sound pressure at Outlet k can be calculated using transfer matrix between Inlet j and Outlet k and the circuit analogy source model. Hence,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
The load sound pressure and particle velocity can be expressed as
[p.sub.Lj] = [p.sub.i,j] + [p.sub.r,j] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
where [z.sub.C] is characteristic impedance, and the superscript a indicates the transfer matrix is obtained with anechoic boundary conditions applied to all inlets and outlets other than Inlet j and Outlet k. From the circuit analogy in Fig. 1,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)
Since outlets are assumed anechoic,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)
The transfer function between transmitted sound pressure at Outlet k and Source Strength j can be calculated from Eqns. (4), (5), (6), (7), (8). Hence,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
With all sources active, the sound pressure at each outlet can be calculated by summing the contribution from each source. The amplitude and phase relationship between sources can be described using complex ratios [alpha] In this work, the reference source is chosen to be the source at Inlet 1.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Plugging in Eqns. (9), (10), (11), Eqn. (3) can be simplified as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)
The insertion loss of a MIMO muffer is defined as the ratio between the summation of transmitted sound power in each outlet to the summation of transmitted power if all sources are connected to straight tubes of a certain length. This can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
A similar superposition method to that used to determine transmission loss can be used to calculate [p.sub.k] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. [p.sub.k] is the sound pressure in the outlet pipe for the case with muffer (Fig. 4) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the sound pressure at the outlet for a straight pipe (Fig. 5). The difference in this calculation is that realistic source and termination impedances are applied as boundary conditions.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)
where the superscript Z indicates the transfer matrix is obtained with realistic impedance boundary conditions applied on all inlets and outlets other than Inlet j and Outlet k. The source and termination impedances can be expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
The transfer function between transmitted sound pressure at Outlet k and Source j can be calculated from Eqns. (14), (15), (16) and is written as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
The transfer functions for straight tube connections can be derived in the same way and are expressed as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)
where the superscript t indicates the transfer matrix entries are for a straight tube. Plugging in Eqns. (17)-(18), Eqn. (13) can be simplified as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (19)
To validate the superposition method, a 2-inlet 2-outlet muffer is built (Fig. 6). The muffer cylinder has a length of 25 inches and diameter of 10 inches. Plates of lengths 10 and 8 inches are inserted to add complexity and avoid symmetry. The test setup is shown in Fig. 7. Two compression drivers (JBL 2447H and 2426H) are used as sources. The compression drivers are connected to the inlets using Spectronics impedance tubes. To conveniently control the phase difference between these two sources, sine waves are used as driving signals. The central frequencies of octave bands from 125 Hz to 4000 Hz are selected. At each frequency, the phase delay between Source 1 and Source 2 is changed from 0 to 180 degrees, with step sizes of 45 degrees. The sound pressure at each outlet is measured and compared against the prediction determined using the superposition model.
The source strengths and source impedances of both compression drivers are measured at specified frequencies using the multi-load method . The four loads used in determining the source properties are a simple expansion chamber, straight tube, divergent cone, and foam termination. The measured source strengths are phase-referenced to the input signal. The measured source strengths and source impedances for both compression drivers are shown in Fig. 8 and 9.
Additionally, termination impedance is needed to predict the sound pressure at the outlets. The test point is selected to be 8 inches from the opening of the outlets. The impedance at this point towards the opening is measured using ASTM-E1050 , and the measured termination impedance is shown in Fig.10.
To predict the sound pressure at the outlets, the transfer matrices between inlets and outlets must be measured with realistic boundary conditions applied at the ports. The transfer matrices are measured using ASTM-E2611 . To keep the boundary conditions unchanged, when measuring transfer matrix between one inlet and one outlet, the compression driver at the other inlet is still active but with a signal about 1/100 of the normal amplitude. This small amplitude is shown to excite the compression driver to a minimal source strength and preserve the source impedance .
RESULTS AND DISCUSSION
The sound pressures at the outlets, both directly measured and predicted using the superposition model, are shown in Fig. 11 and 12. From the comparisons, it can be seen that the prediction using the superposition method is very accurate for a variation of phase delays.
After the superposition model was validated, the transmission loss and insertion loss of the muffer can be calculated for different phase delays between the sources. To calculate transmission loss, transfer matrices with all other ports anechoic are required. These transfer matrices can be obtained using simulation or approximated using measurement. In current work, a measurement method is used. When transfer matrices are measured, the unused ports are closed with 10-inch foam (Fig. 13), which can be considered as approximately anechoic. The test setup and absorption coefficients of these two foams are shown in Fig. 14. The calculated insertion loss and transmission loss are shown in Fig. 15 and 16 respectively.
The trend of insertion loss variation with phase delay increase correlates well with the results for outlet sound pressure. From Fig. 11 and 12, it can be seem that below 1000 Hz, the sound pressures at both outlets increase with phase delay increases. In this frequency range, the insertion loss decreases with increasing phase delay. At 1000 Hz, the influence of phase on the outlet sound pressure is negligible for both outlets, and insertion loss remains constant with varying phase delay. At the frequency of 2000 Hz, with phase delay increases, the sound pressure decreases at Outlet 1 while increasing at Outlet 2. The insertion loss remains constant suggesting that the corresponding increase and decrease of outlet sound pressure counteract one another.
The transmission loss calculated using the superposition model also shows a very similar trend compared to insertion loss. It demonstrates that without knowledge of source and termination impedance, transmission loss provides an estimate of the actual performance. It is shown that phase delay plays a more important role in the lower frequency range than at higher frequencies for this 2-inlet 2-outlet muffer. Though only one example is shown in this paper, similar conclusions were also seen for a more practical muffer .
Compared to the analytical mode-matching and impedance matrix approaches, the advantage of the superposition approach is that the measurement and simulation techniques to obtain transfer matrices are very well developed, and the concept of superposition is mathematically straightforward to understand.
In this paper, definitions of transmission loss and insertion loss are given for MIMO muffers based on a superposition method. The approach was validated experimentally on a 2-inlet 2-outlet muffer. It is observed that low frequency performance of a muffer is more sensitive to the phase difference between sources.
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[12.] Hua, X., Jiang. C., Herrin, D.W., and Wu, T.W., "Determination of transmission and insertion loss for multi-inlet mufflers using impedance matrix and superposition approaches with comparisons." Journal of Sound and Vibration333, no. 22 (2014): 5680-5692.
Yitian Zhang, David W. Herrin, and T. Wu
University of Kentucky
Faurecia Emissions Control Technologies
D. W. Herrin
Department of Mechanical Engineering
University of Kentucky
Lexington, KY 40506, USA
The authors gratefully acknowledge the support of the Vibro-Acoustics Consortium at University of Kentucky.
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|Author:||Zhang, Yitian; Herrin, David W.; Wu, T.; Hua, Xin|
|Publication:||SAE International Journal of Passenger Cars - Mechanical Systems|
|Date:||Apr 1, 2016|
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