Active control of energy density in a 1D waveguide: A cautionary note Running Title: Energy density control in a duct Ben S. Cazzolato, Dick Petersen,† Carl Q. Howard,‡ and Anthony C. Zander§ School of Mechanical Engineering, The University of Adelaide, SA, 5005, AUSTRALIA.
Abstract Acoustic energy density has been shown to be a highly effective cost function for active noise control systems. Many researchers have used the sound field in a one-dimensional waveguide to trial their control strategies before moving onto more realistic three-dimensional sound fields. This letter aims to shed some light on the observations made in the early papers on one-dimensional energy density control and also shows that some of the analysis was incorrect and the conclusions reached may be flawed. PACS numbers: 43.50.Ki
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I.
INTRODUCTION
The popularity of energy density sensors for use in active noise control systems has grown strongly in recent years, with investigations on both one-dimensional systems [1–9] and threedimensional systems [10–19]. Acoustic energy density, defined as the sum of the acoustic potential and kinetic energy densities at a point, is commonly estimated using the two-microphone method [1] and is given by (see Appendix A) ED z
p2 z 2ρc2
ρv2 z 2
(1)
where z is the location, ρ is the density of the fluid and c is the speed of sound in the fluid. The mean of the outputs from the two microphones provides an estimate of the acoustic pressure p, the square of which when normalised by 1 2ρc2 , is equal to the acoustic potential energy density at a point. The pressure difference between the two microphones is used to calculate the acoustic particle velocity v, the square of which when normalised by ρ 2, is equal to the acoustic kinetic energy density at a point. Researchers investigating active noise control have often used the sound fields inside onedimensional waveguides to investigate various control strategies prior to moving to realistic three-dimensional sound fields. The research into active noise control using energy density as a cost function is no exception. Reviews of the current literature show two main energy density control strategies in one-dimensional reactive fields; global control using a single control source [1–4] and local control using two control sources [5–8]. For all these cases the onedimensional sound field has been modelled using a rigid walled one-dimensional modal model. It will be shown that this modelling technique has obscured the true results and often misled researchers into believing their results to be accurate. The work presented here has used a travelling wave solution as an alternative formulation. This solution can be shown to coincide to the modal solution for rigid-walled rectangular cavities [20]. Solving this algebraically, as opposed to past research which used numerical solutions of the modal formulation, it will be seen that all research to date has been severely compromised by numerical noise. The added advantage of the travelling wave model is the computational efficiency in which it may be used for any arbitrary termination condition, unlike the modal solution which may require a very large number of modes for non-rigid terminations. 2
II.
MODELLING THE ACOUSTIC RESPONSE OF A RIGID-WALLED FINITE DUCT
Consider the hard-walled one-dimensional waveguide of finite length L and arbitrary termination conditions described by Φ1 and Φ2 as shown in Figure 1. The pressure response in the duct at a point z arising from a source with unit volume velocity located at z s is given by the summation of all possible direct and reflected wave components [21, 22] ρcT e 2S
p z zs
jk zs z
e
jk zs z
e
2Φ1
jk zs e
z
e
2 jkL 2Φ2
jk zs z e e
2 jkL 2Φ1 2Φ2
(2) where k is the acoustic wavenumber, c is the speed of sound, ρ is the density of air, S is the duct cross section, assumed to be small relative to the wavelength such that only plane waves propagate, and T is the modal reverberation factor given by T
1
1 e 2 jkL e
where the termination phasors Φ1 2 πα1 2
by R1 2 e
2Φ1 2
2Φ1 e 2Φ2
(3)
jπβ1 2 [23] are related to the reflection coefficients
[23], where the exponential form results in a compact expression for Equations
(2) and (3). The appropriate values for α and β yield specific termination conditions. For example, a rigid, totally reflective termination is achieved by setting α β 0 (Φ 0), while an anechoic termination is achieved by making the real part of Φ infinite, namely α ∞. Looking at Equation (2) it can be seen that the sound field is made up of two travelling waves; a forward travelling wave (first two terms) and backward travelling wave (second two terms). The infinite number of reflections from the two end caps is accounted for by the modal reverberation factor in Equation (3). It should be noted that it has been assumed that the sound field is entirely one-dimensional and that evanescent propagation of cross-modes has not been included. Consequently there exists no near field effects adjacent to sources.
III.
ENERGY DENSITY CONTROL
The travelling wave formulation will now be applied to two scenarios modelled previously; global control with one control source and local control using two control sources. Like previous work, causality has been neglected when formulating the control law. This is acceptable 3
when modelling a feedforward control strategy such as the FX-LMS algorithm. The optimal strength of the control source(s), qs , is found by minimising the following Hermitian quadratic form cost function [24] J
qHs Aqs qHs b bH qs c
(4)
where the Hermitian control matrix A ZH Z, the vector b ZH p p and the scalar c
pHp p p .
The latter is the (weighted [17]) sum of the squared pressures at the error sensor locations for the primary field, p p . This weighting is such that energy density is minimised. The individual terms of the control source transfer function matrix Z are given by Equation (2). Provided that the control matrix A is positive definite and full-rank, the optimal control source strengths and corresponding minimum value of the cost function are given by [24] qs opt
A 1b
Jopt
c bH A 1 b
(5)
A. One control source
Consider the case of a single control source investigated in Refs [1–4], where the energy density was estimated using the two-microphone technique [1] (which is briefly summarised in Appendix A). Using simulations based on a modal model previous researchers found that when the source was located upstream of the two microphones (z s
zm ) the sound pressure
levels downstream of the source were reduced by between 20 and 40dB. Their experiments show levels of attenuation at least 10dB higher than predicted, whereas simulations typically provide an estimate of the maximum levels of attenuation one could expect to achieve. The reason for this discrepancy can be found by considering the travelling wave model. It can be shown that the solution to minimising the pressure at two microphones downstream of the source results in complete attenuation of pressure at all locations downstream of the control source [24]. It therefore follows that the necessary relationship between the primary source of strength q p located at z 0 and the secondary source at zs is qs
qp
e
jkzs
e
1
2 jkzs
e2Φ1 e2Φ1
which converges to the solution obtained for an infinite waveguide as ℜ Φ 1
4
(6) ∞, namely
qs
qpe
jkzs .
The pressure response after control is therefore given by p z zs
ρc e 2S
jkz
e2 jkz e2 jkzs 1 e2Φ1 1 e2 jkzs 2Φ1
z zs
(7)
0
z zs
The net effect of the control source is to completely reflect the sound back upstream via a pressure release boundary condition [24]. The reason that this solution was not achieved in the previous work is due to small phase errors in the pressure response estimates. Modal truncation is the source of these errors in the numerical modal model. In fact, as the total number of modes in the formulation is increased, the solution converges to the travelling wave solution up to a point, as illustrated in Figure 2. Eventually the numerical precision of the computer program is reached which then introduces noise and prevents any further convergence. This has been observed by the current authors. The noise in the previous experiments may have come from many sources [25] such as resolution bandwidth errors, poor coherence, etc. The closed circular duct used in Refs [1–4] has been used to illustrate these results and issues. Figure 2 shows the pressure response of a duct L
5 6m in length and 0 116m in
diameter, driven at a frequency of 200Hz by a primary source positioned at one end of the duct and a control source located arbitrarily at 0 34L. The speed of sound c
343m s and
the density was ρ 1 2kg m3 . The modal quality factor of Q 50 was used, corresponding to termination phasors of approximately Φ1
Φ2
1 9.
Two microphones separated by 1 7cm
with a mid-point located at 0 47L were used as error sensors. The results from the travelling wave solution are compared with results from a modal model using various numbers of modes. The primary sound field for all solutions is almost identical, as is the controlled sound field upstream of the control source. However, the controlled pressure downstream of the control source is quite different, even between modal solutions. Note that the controlled sound field of the modal model begins to converge to the travelling wave solution as more modes are added. Eventually the modal model ceases to change with the addition of modes. It was found that this is due to numerical noise associated with the double precision calculations.
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B. Two control sources
Since it was shown in Section III A that only a single upstream control source is required to attenuate completely the acoustic energy density downstream of the control source, it is clear that two control sources upstream of the two microphones will result in a rank deficient control matrix. Kestell et al [5–8] investigated the case of two control sources downstream of the two microphones. It will be shown in the following analysis that this is a poorly posed problem and that the control matrix is rank deficient. The transfer function between two microphones located at zm1 and zm2 when driven by a source located at zs in the duct is given by e
H12
1 e2 jkzm2 2Φ1 1 e2 jkzm1
jk zm1 zm2
e
0 z m1
2 jkzm2 2 jkzs 2Φ1 2 jkL 2Φ2 jk zm 1 zme 2 jkz me 2Φ1 e2 jkzs 2 jkL 1 1 2 1 e e e 2 jkzm2 2 jkL 2Φ2 e e e jk zm1 zm2 e2 jkzm1 e2 jkL 2Φ2
zm2
zs
L
2Φ1
2Φ2
0 z m1
0 zs
zm1
zs zm2
zm2
L
(8)
L
From Equation (8) it can be seen that when the source is not positioned between the two microphones, the transfer function between the two microphones is independent of the source position. This is to be expected and is the basis for the two-microphone transfer function method used to measure impedance and absorption in impedance tubes [26]. In fact, solving Equation (8) for the complex reflection coefficient when the source is upstream of the two microphones gives R2 where s zm2
e
2Φ2
e2 jk
L zm 1
H12 e
jks
e jks
H12
(9)
zm1 is the microphone separation. This is the same expression derived by Chung
and Blaser [26]. Given that the transfer function between the two microphones is independent of source position (zs ) when the source is located downstream (or upstream) of both microphones, then it is obvious that the second control source does not offer any more control authority than a single control source. Therefore two control sources located downstream (or upstream) of an energy density sensor is a rank deficient problem. This implies that both the numerical and experi-
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mental work of Kestell et al [5–8] is incorrect and explains why they had such poor correlation between their numerical simulations and experimental results. Presumably the reason that they were able to calculate the strengths of the control sources at all was due to the same reasons discussed in the previous section. Noise in both the numerical and experimental results meant that the condition number of the control matrix was no longer infinite and therefore invertible. However this matrix would have been extremely poorly conditioned, indicative of a poorly posed control problem. This has in fact been observed in simulations using modal models conducted by the current authors.
IV. CONCLUSIONS
It has been shown that for a single control source in a finite rigid-walled one-dimensional waveguide it is possible to achieve infinite attenuation of energy density when the primary and control source is located upstream of the energy density sensor. This explains the discrepancy Sommerfeldt et al [1–4] observed between numerical and experimental results. It has also been shown that adding a second control source downstream of the sensor results in a rank deficient control matrix. This implies that all the previous simulations and experiments investigating the zone of local control using energy density sensing by Kestell et al [5–8] are incorrect, and the conclusions drawn from these results may be flawed. It is perhaps worth reinforcing that the results presented here are not exclusively limited to 1D waveguides and that results for 3D simulations also need to be checked for modal convergence.
Acknowledgments
The financial support of the Australian Research Council is gratefully acknowledged.
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[1] P.J. Nashif and S.D. Sommerfeldt. An active control strategy for minimising the energy density in enclosures. In Proceedings of Inter Noise 92, pages 357–361, 1992. [2] S.D. Sommerfeldt and P.J. Nashif. Energy based control of the sound field in enclosures. In The Second International Congress on Recent Developments is Air- and Structure-Borne Sound and Vibration, pages 361–368, 1992. [3] S.D. Sommerfeldt and P.J. Nashif. An adaptive filtered-x algorithm for energy based active control. Journal of the Acoustical Society of America, 96(1):300–306, 1994. [4] Y.C. Park and S.D. Sommerfeldt. Global attenuation of broadband noise fields using energy density control. Journal of the Acoustical Society of America, 101(1):350–359, 1997. [5] C.D. Kestell, B.S. Cazzolato, and C.H. Hansen. Virtual energy density sensing in active noise control systems. In Proceedings of the International Congress on Sound and Vibration, 2000. [6] C.D. Kestell, C.H. Hansen, and B.S. Cazzolato. Active noise control with virtual sensors in a long narrow duct. International Journal of Acoustics and Vibration, 5(2):63–76, 2000. [7] C.D. Kestell, C.H. Hansen, and B.S. Cazzolato. Virtual sensors in active noise control. Acoustics Australia, 29(2):57–61, 2001. [8] C.D. Kestell and C.H. Hansen. Active noise control with virtual sensors. In Proceedings of the International Congress on Sound and Vibration, 2001. [9] B.S. Cazzolato and C.H. Hansen. Errors in the measurement of acoustic energy density in onedimensional sound fields. Journal of Sound and Vibration, 236(5):801–831, 2000. [10] S.D. Sommerfeldt, J.W. Parkins, and Y.C. Park. Global active noise control in rectangular enclosures. In Proceedings of Active 95, pages 477–488, 1995. [11] S.J. Elliott and J. Garcia-Bonito. Active cancellation of pressure and pressure gradient in a diffuse sound field. Journal of Sound and Vibration, 186(4):696–704, 1995. [12] S.D. Sommerfeldt, J.W. Parkins, and J. Tichy. Modal results for active control of energy density in a rectangular enclosure. In Proceedings of Noise-Con 96, pages 429–434, 1996. [13] W. Shen and J.Q. Sun. A study of shell interior noise control. SPIE, 3041:812–818, 1997. [14] J.W. Parkins, S.D. Sommerfeldt, and J. Tichy. Error analysis of a practical energy density sensor.
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Journal of the Acoustical Society of America, 108(1):211–222, 2000. [15] J.W. Parkins, S.D. Sommerfeldt, and J. Tichy. Narrowband and broadband active control in an enclosure using the acoustic energy density. Journal of the Acoustical Society of America, 108(1): 192–203, 2000. [16] S.K. Lau and S.K. Tang. Sound fields in a slightly damped rectangular enclosure under active control. Journal of Sound and Vibration, 238(4):637–660, 2000. [17] B.S. Cazzolato and C.H. Hansen. Errors arising from three-dimensional acoustic energy density sensing in one-dimensional sound fields. Journal of Sound and Vibration, 236(3):375–400, 2000. [18] S.K. Lau and S.K. Tang. Impacts of structural-acoustic coupling on the performance of energy density-based active sound transmission control. Journal of Sound and Vibration, 266(1):147–170, 2003. [19] B.S. Cazzolato and C.H. Hansen. Active control of enclosed sound fields using three-axis energy density sensors: Rigid walled enclosures. International Journal of Acoustics and Vibration, 8(1): 39–51, March 2003. [20] D. Lubman. Equivalence of eigenmode and free-wave models of steady-state reverberation in rectangular rooms. Journal of the Acoustical Society of America, 60(Supp 1):S59, 1976. [21] A.C. Zander and C.H. Hansen. A comparison of error sensor strategies for the active control of duct noise. Journal of the Acoustical Society of America, 94(2):841–848, 1993. [22] S.J. Elliott and P.A. Nelson. Models for describing active noise control in ducts. ISVR Technical Report 127, ISVR, University of Southampton, 1984. [23] P.M. Morse and K.U. Ingard. Theoretical Acoustics. McGraw-Hill Book Company, New York, 1968. [24] P.A. Nelson and S.J. Elliott. Active Control of Sound. Academic Press, London, 1992. [25] J.S. Bendat and A.G. Piersol. Random Data - Analysis and Measurement Procedures. John Wiley & Sons, New York, 2nd edition, 1986. [26] J.Y. Chung and D.A. Blaser. Transfer function method of measuring in-duct acoustic properties. I. Theory. Journal of the Acoustical Society of America, 68(3):907–913, 1980.
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Appendix A: ENERGY DENSITY CONTROL
The instantaneous acoustic energy density at a point z is given by [1] p2 z 2ρc2
ED z
ρv2 z 2
(A1)
where p and v are the instantaneous acoustic pressure and particle velocity at z respectively. The particle velocity is related to the pressure gradient by the following v z
∂p z dt ∂z
1 ρ
(A2)
The spatial derivative of the pressure is commonly estimated using two closely spaced microphones to provide a first order spatial derivative given by ∂p z ∂z
p2
p1
(A3)
2h
where p1 and p2 are the pressures at the two microphones separated by a distance of 2h. Evaluating the temporal integral in Equation (A2) and substituting the pressure gradient estimate given by Equation (A3), the particle velocity estimate can be rewritten in terms of the acoustic wavenumber k ω c , and is given by v z
j p2 p1 ρω 2h
j p2 p1 ρck 2h
(A4)
Using the average of the two microphone signals as the pressure estimate, the instantaneous acoustic energy density at the mid-point between points 1 and 2 is given by Eˆ D z
1 p1 p2 2 8ρc2
10
p2
kh
p1 2 2
(A5)
LIST OF FIGURES
1
Schematic of the 1D system under investigation. The duct is length L with arbitrary termination conditions described by the complex phasors Φ 1 and Φ2 . There is a volume velocity source at a distance zs from end Φ1 and two microphones at zm1 and zm2 .
2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
Pressure response in a duct with a single control source (zs ) located upstream of two microphones (zm ). The primary source was located at one end of the duct (z p ). The black lines indicate the primary sound field and the grey lines indicate the controlled field. The solid lines have been generated using the modal model and the dashed lines are using the travelling wave model. The parameter N represents the number of axial modes included in the modal model. . . . . . . .
11
13
L zs Loudspeaker
F1
z
Microphones
Arbitrary F 2 Termination
zm1 zm2 Figure 1: Schematic of the 1D system under investigation. The duct is length L with arbitrary termination conditions described by the complex phasors Φ 1 and Φ2 . There is a volume velocity source at a distance zs from end Φ1 and two microphones at zm1 and zm2 .
12
Energy density control in duct
Acoustic pressure amplitude → dB re 20 µ Pa
100
50
0
−50 −100 ← z
←z
p
s
−150
←z
m
Primary field Controlled field N = 50 Controlled field N = 200 Controlled field N = 1000 Primary field Controlled field
−200
−250 0
0.1
0.2
0.3 0.4 0.5 0.6 0.7 Normalised duct length → z/L
0.8
0.9
1
Figure 2: Pressure response in a duct with a single control source (z s ) located upstream of two microphones (zm ). The primary source was located at one end of the duct (z p ). The black lines indicate the primary sound field and the grey lines indicate the controlled field. The solid lines have been generated using the modal model and the dashed lines are using the travelling wave model. The parameter N represents the number of axial modes included in the modal model.
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