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Actuators positioning for multichannel active control system in circular ducts
Applied Acoustics 59 (2000) 323±335 www.elsevier.com/locate/apacoust
Actuators positioning for multichannel active control system in circular ducts Alex Boudreau a,*, Andre L'EspeÂrance a, Martin Bouchard b, Bruno Paillard a a G.A.U.S. University of Sherbrooke, Sherbrooke, QueÂbec J1K 2R1, Canada School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada
b
Received 8 February 1999; accepted 21 June 1999
Abstract In the particular case of multichannel control, the acoustic power limit of loudspeakers may be an important restriction for active control application when the noise levels are very high. The goal of this paper is to present the dierent parameters that allow to minimize the power used by the control sources for a multichannel active control system in a circular duct. An experimental study of the longitudinal distribution pressure ®eld for the particular case of higher order modes, as well as some experimentations with active control for dierent geometric conditions have been done to analyze this problem. The results of these experimentations have allowed us to understand that the most in¯uential phenomenon that determines the control sources optimum position are the re¯ection from ducts closed end. With this result in mind, a simple and ecient methodology of positioning has been developed. The eciency of this positioning method has been proved from experimental tests. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The positioning of the error sensors for a multichannel control system in a circular duct is currently an important subject of research [1±4]. Indeed, the sensors positioning plays a decisive role on the noise reduction obtained. However, the control sources positioning is still a problem, and it is almost absent of the literature. Even
* Corresponding author. E-mail address: [email protected] (A. Boudreau). 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00034-1
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so, the use of several speakers for a multichannel control system presents an important problem regarding the necessary acoustic power needed for each speaker. In fact, the necessary power for the control sources can occasionally become greater than the one generated by the primary source, depending on the transfer function combination between dierent speakers and error microphones. The required power for the control sources is, sometimes, an important limitation for industrial applications, mostly when the primary acoustic ®eld is very high. Thus it seems that the optimization of the control sources positioning is an important factor, in order to minimize the power used. One of the objectives of this paper is to quantify the importance of the limited power for various parameters: frequency, ducts length, control sources positioning and error sensors plane. The ®rst section will present an experimental analysis of the distribution pressure ®eld in a semi-open duct. The eect of the source position and of the duct length on the interference pattern will be analyzed from an active control point of view. The second section will study an analysis of the acoustic power used by actuators in a multichannel system case. The eects of closed-end re¯ections, actuators position and error sensors position on the power used for the control sources will be analyzed. With this in mind, a positioning method has been developed and evaluated by experimental tests to minimize the power used by control actuators. 2. Longitudinal distribution of acoustic pressure ®eld In the case of the inside duct sound propagation, it is common knowledge that the stationary waves are established to form an interference pattern, following the re¯ection coecients and radiation impedance at the duct end [4±7]. Understanding this phenomenon of propagation is relatively simple for low frequencies when the plane mode (0,0) is the only one to be propagated in the duct. However, when several propagation modes occur, the inside duct pressure ®eld distribution becomes more complex because of the modes combination. In those cases, the pressure ®eld is not uniform, not only following the duct axis, but also in each duct section. Before analyzing the power actuators problem for the multichannel active control, it appears important to review the inside duct pressure ®eld longitudinal distribution for the higher order modes. 2.1. Experimental assembly To study the inside duct pressure ®eld longitudinal distribution, an experimental study has been realized with the assembly shown in Fig. 1. The pressure ®eld is generated in a PVC duct of 30 cm diameter and 2.05 m in length, with a closed and open end. A ®xed speaker on the duct wall is used to generate a pressure ®eld in the duct. The measurements were taken with the help of an acquisition system which uses a ®ve rotary microphones probe. A rod following the duct length axis is used to rotate the ®ve microphones, which are placed on a support in the duct radius axis. These ®ve microphones are distributed in uniform way, along the radius. A variable
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step electric motor is used to rotate the rod and to move the ®ve microphones, at every 10 . A multiplexer is installed to make the transition from one microphone to another. This assembly allows to choose the measurement plane according to the duct longitudinal axis. A white noise between 100 Hz and 2 kHz is generated by the speaker installed on the duct wall. The noise level recordings have been done using a program on a PC. This program records 801 lines of an autospectrum, with the HPIB port of an analyzer (BK2034). For the studied frequency range (100 Hz to 2 kHz), some higher order propagation modes are found. The cut-o frequency of the ®rst mode for the dimensions of the duct is 675 Hz. For the case of the frequencies higher than 675 Hz, the pressure ®eld in a section of the duct is not uniform. The measurements for each microphone allow to draw the noise level distribution of one frequency for a given duct section. For instance, Fig. 2a and b gives, respectively, the pressure ®eld distribution for a section at 800 and 1300 Hz. In Fig. 2a, it is easy to recognize the creation of a nodal line following the duct diameter, since the mode (0,0) and (0,1) are the only ones to be propagated. The combination of modes (0,0), (0,1) and (0,2) at 1300 Hz generates
Fig. 1. Measurement of acoustic pressure ®eld in a section: experimental assembly.
Fig. 2. (a) Distribution of noise levels in a section at 800 Hz, and (b) 1300 Hz.
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a complex pressure ®eld distribution that makes it dicult to identify the nodal lines. Since it is dicult to study the longitudinal distribution of the noise levels from a section to another, it seems preferable to use the measured mean level, in one section, as an indicator. 2.2. Measurements results 2.2.1. Overview Fig. 3a±d present examples of mean noise level distribution following the duct axis at 800, 1000, 1200 and 1400 Hz, respectively. The global mean level inside the duct varies signi®cantly with the frequency. These results show the interference pattern following the duct axis. However, these patterns become lower for high frequencies. Indeed, the dierence between maximum and minimum mean level goes from 14 to 4 dB between 800 and 1400 Hz. 2.2.2. Eect of the actuator position on the interference pattern The source position following the duct axis has an eect on the interference pattern identi®ed before. To illustrate this phenomenon, two positioning con®gurations for the noise source are presented (see Fig. 4). Because the interference patterns are well de®ned at low frequencies, a frequency of 800 Hz will be used to describe the eect of the source position. Fig. 5 shows the results of the two positions from Fig. 4. The noise source position following the duct axis has no eect on the interference pattern position and shape. On the other hand, the mean pressure level for the ®rst con®guration is approximately 5 dB higher than the one from the second con®guration. On primary analysis, we understand that this general diminution of the pressure ®eld results from the phase between the direct and re¯ected waves for the ®rst mode on the duct closed end (see Fig. 6). For a phase of approximately 180 , the progressive wave pressure is minimized, even though a phase of 0 or 360 increases that pressure. For higher frequencies, upper modes appear and the relative eect of these variations (associated to the plane mode) are reduced. Thus, at low frequencies, when a small number of modes are propagated, and the predominance of the plane mode continues to be important, we see that the source position plays a signi®cant role on the global mean level inside the duct. For the ®rst con®guration, the phase between emitted and re¯ected wave is ÿ14.5 . The second con®guration shows a phase of 140.5 . Thus, we can conclude that the con®guration #1 is a more ecient source position. 2.2.3. Eect of the duct length on the interference pattern The eect of the duct length on the interference pattern position and its shape has been evaluated with the help of a third con®guration shown on Fig. 7. The length of the duct has been reduced from 2.05 to 1.85 m in comparison to the one from con®gurations #1 and #2. The measurements of the mean levels are taken at the same place than the ones for the ®rst and the second con®guration. The closed end of the duct is used as a reference.
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Fig. 3. (a) Interference pattern versus duct axis at 800 Hz; (b) 1000 Hz; (c) 1200 Hz; and (d) 1400 Hz.
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Fig. 4. Con®gurations used for the primary source.
Fig. 5. Mean levels for case #1 and #2 used for primary source.
Fig. 6. Schematic explanation of the re¯ex wave's eect on the global noise level inside the duct.
Fig. 7. Con®guration #3.
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Fig. 8 shows the mean levels for the three con®gurations, for dierent measurement locations. From the results of the previous three subsections, the following conclusions have been made: a. The re¯ection of the plane mode from the duct closed end has a signi®cant eect on the pressure ®eld mean level inside the duct. b. The interference pattern position is not in¯uenced by the source position but only by the duct length. c. Finally, the amplitude of the interference pattern pressure ®eld inside the duct decreases at high frequencies. 2.3. Theoretical simulation The same kind of results have been obtained for the last three con®gurations with a theoretical model of pressure ®eld in a circular duct [4]. Fig. 9 shows results from the simulations for three con®gurations. Although the relation between the theoretical and experimental results are not perfect,1 the general tendencies are well respected. 3. Active control experimentation The results from the previous sections have shown that the noise source position governs the interference pattern amplitude, particularly at low frequencies This section will consider that problem, in the case of the multichannel active control. In the beginning, the plane mode re¯ection problems will be studied, with the help of control experimentation. The eect of the error sensors and the control sources plane position will be considered later.
Fig. 8. Mean levels for cases #1, #2 and #3 used for primary source. 1 The dierences between the theoretical and experimental results can be due, in part, to the interaction between the ¯uid and the assembly structure. Indeed, in comparison to the theoretical model with the rigid and perfect re¯ective wall assumptions, the assembly vibrates and transmits noise.
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3.1. Control problems in presence of re¯ections When the position of the control sources produces a phase of 180 between the emitted wave and re¯ected wave from the duct closed end, the power used by the control sources becomes more important. In order to explain this phenomenon, some control experimentations with and without re¯ections have been realized using the assembly shown in Fig. 10. To make these control experimentations, a feed-forward multichannel system has been used. This control algorithm running on a DSP C31 is linked to a PC board [8]. All the control speakers have the same position following the duct axis and are distributed in uniform way around the duct circumference. The active control experimentation will cover a frequency range from 1000 to 1350 Hz. For that interval, ®ve microphones and ®ve speakers are used. The number of channels used and the microphones disposition in the control plane are based on the methodology described in Ref. [4]. The control experimentations have been made using a pure tone with a frequency step of 10 Hz. The power used by the control sources has been estimated for each frequency. To study the eect of the re¯ections, two cases have been considered: the ®rst case with and the second one without absorbent material covering the duct closed end. For these control experimentations, the error microphones plane A has been used (see Fig. 10). The normalized electric power used by the control sources is calculated
Fig. 9. Mean levels for cases #1, #2 and #3 used for primary source (model).
Fig. 10. Assembly used for control experimentation.
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from rms voltage measures. These measurements are taken from each control source and are normalized by the primary source voltage: P
2rms sc
2rms sb
where: sb=noise source used to generate the primary ®eld sc=control sources. Fig. 11a and b give the normalized electric power used by each control source for the cases with and without absorbent material. For the normalized power higher than unity, the control sources require more acoustic energy than the primary source. Then, we have an overconsumption problem. These results show that the re¯ections from the duct closed end have an important eect on the power used by the control sources. Furthermore, the graph on Fig. 11 shows that the overconsumption problem is more important at low frequencies, and that it becomes blurred for higher frequencies because the plane mode interference becomes less and less important. 3.2. Eect of the sensor plane position Due to the interference pattern following the duct axis, the microphone plane can be on a maximum or a minimum of pressure (see Section 2.2). To check how the microphone plane's position on the interference pattern in¯uences the control sources consumption, a second microphone plane position (B) has been considered (see Fig. 10). The control experimentation has been done at 800 Hz. The graph on Fig. 12 shows the normalized power for the two microphone planes' positioning. The last graph does not show a lot of variations for the control sources used power between the two error sensors positioning. The mean noise reduction measurements at the open end with and without control show that the eciency is smaller when the control is applied on a low noise level zone (noise reductions: 31.1 versus 39.3 dB). This problem is related to the dynamic range of the input signal acquisition system. In a minimum pressure location, the microphones become more rapidly in measurement noise and, in this way, the control is less ecient. These results bring to the conclusion that the control sources used power reduction with the help of the error sensor plane positioning seem to be impossible. 3.3. Eect of the control sources' position The phase between the direct and re¯ected wave from the duct closed end seems to be the most in¯uential parameter for the control sources eciency. If the positioning of the control sources is in such a way that the phase is close to 0 , it seems possible to minimize the used power.
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Fig. 11. (a) Electrical power used versus frequency for a ®ve channel active control system in presence of re¯ections from the closed end. (b) Electrical power used versus frequency for a ®ve channel active control system without the presence of re¯ections from the closed end.
However, it is not practical, on the experimental assembly, to change the primary and control sources position. The method used for this paper consists in doing a study of the phase between the emitted and re¯ected wave from the duct closed end versus the frequencies for the con®guration showed in Fig. 10. Fig. 13 shows the phase versus the frequencies for two speakers planes and for the geometric conditions of Fig 10. To make easier the analysis of the frequencies range for which the sources positions are optimum, the graph of Fig. 14 has been built.
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Fig. 12. Electrical power used by each sources for sensor plane's position A and B.
Fig. 13. Phase between the primary source and the control sources versus frequency.
Fig. 14. Dierence of phase versus frequency.
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Fig. 15. Electrical power used by the control sources at 800 and 920 Hz.
This last graph shows the dierence between the absolute value of the plane mode phase for the noise source and the control sources. It is easy to judge which kind of source is more ecient. The positive zones show the frequencies where the control sources eciency is predominant, and the negative zones indicate that the noise source is more eciency. At 920 Hz, the control sources used power is smaller compared to 800 Hz. In fact, Fig. 14 shows a better eciency for the noise source at 800 Hz. To con®rm that the control sources used power is lower at 920 Hz, a control experimentation has been done. The used geometric con®guration is showed in Section 3.2. The results of the control experimentation are showed in Fig. 15. This graph shows the normalized power for 920 and 800 Hz (for microphone plane A, see Fig. 10). The noise reduction is practically the same at 920 and 800 Hz but, at 920 Hz, the control sources used power is lower. The maximum normalized power at 920 Hz is less than 0.20. For these two frequencies, two modes are being propagated and the pressure ®eld complexity is similar. 4. Conclusion For a multichannel control system, the control sources used power may become a real limitation problem. To analyze this problem, an assembly has been realized in laboratory. This assembly allows to analyze the pressure ®eld longitudinal distribution inside the duct. Many multichannel control experimentations for various frequencies, control sources positions and error sensors plane positions have been realized. The results of these experimentations show that the mean pressure ®eld longitudinal distribution has an important variation, and that the interference pattern position depends on the duct length and not on the noise source position. These variations are generated by the plane mode interference. This phenomenon becomes
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less important at high frequencies because the plane mode importance becomes smaller in comparison to the higher order propagation modes. The eciency of a noise source can be modulated by its position following the duct axis. It seems to be possible to optimize the control sources eciency if the plane mode phase between direct and re¯ected wave at the duct closed end is taken into consideration. This control sources optimization method has been checked in laboratory with a multichannel control system, and the obtained results have demonstrated the validity of this approach. Thus, this work allows to install a multichannel active control where the control sources acoustic power is an important constraint. References [1] Baumann DC, Grenier RA. Modal identi®cation approach to multi-modal cancellation. Proc. of Inter-noise 92. p. 341±8. [2] Erikson L. Higher order mode cancellation in ducts using active noise control. Proc. of Inter-noise 89. [3] L'EspeÂrance A, Bouchard M, Paillard B, Guigou C. US patent No. 5, 748, 750±May 5, 1998: Method and apparatus for active noise control of high order modes in ducts. [4] L'EspeÂrance A, Bouchard M, Paillard B, Guigou C, Boudreau A. Active noise control in large circular duct using an error sensors plane. Applied Acoustics 1999;57:357±74. [5] Levine H, Schwinger J. On the radiation of sound from an un¯anged circular pipe. Physical Review 1948;73(4):383±496. [6] Houiel F. ModeÂlisation du champ de pression et simulation de controÃle actif multicanal dans un conduit circulaire. MaõÃtrise es sciences appliqueÂes, speÂcialiteÂ: geÂnie meÂcanique; Groupe d'Acoustique et Vibrations de l'Universite de Sherbrooke, 1995. [7] Homicz GF, Lordi JA. A note on the radiative patterns of duct acoustics modes. J Sound Vib 1975;41(3):283±90. [8] Bouchard M. EÂtude d'algorithmes d'adaptation rapide pour controÃleurs actifs multi-variables. MaõÃtrise es sciences appliqueÂes, speÂcialiteÂ: geÂnie eÂlectrique; Groupe d'Acoustique et Vibrations de l'Universite de Sherbrooke, 1995.
Actuators positioning for multichannel active control system in circular ducts Alex Boudreau a,*, Andre L'EspeÂrance a, Martin Bouchard b, Bruno Paillard a a G.A.U.S. University of Sherbrooke, Sherbrooke, QueÂbec J1K 2R1, Canada School of Information Technology and Engineering, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada
b
Received 8 February 1999; accepted 21 June 1999
Abstract In the particular case of multichannel control, the acoustic power limit of loudspeakers may be an important restriction for active control application when the noise levels are very high. The goal of this paper is to present the dierent parameters that allow to minimize the power used by the control sources for a multichannel active control system in a circular duct. An experimental study of the longitudinal distribution pressure ®eld for the particular case of higher order modes, as well as some experimentations with active control for dierent geometric conditions have been done to analyze this problem. The results of these experimentations have allowed us to understand that the most in¯uential phenomenon that determines the control sources optimum position are the re¯ection from ducts closed end. With this result in mind, a simple and ecient methodology of positioning has been developed. The eciency of this positioning method has been proved from experimental tests. # 2000 Elsevier Science Ltd. All rights reserved.
1. Introduction The positioning of the error sensors for a multichannel control system in a circular duct is currently an important subject of research [1±4]. Indeed, the sensors positioning plays a decisive role on the noise reduction obtained. However, the control sources positioning is still a problem, and it is almost absent of the literature. Even
* Corresponding author. E-mail address: [email protected] (A. Boudreau). 0003-682X/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0003-682X(99)00034-1
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so, the use of several speakers for a multichannel control system presents an important problem regarding the necessary acoustic power needed for each speaker. In fact, the necessary power for the control sources can occasionally become greater than the one generated by the primary source, depending on the transfer function combination between dierent speakers and error microphones. The required power for the control sources is, sometimes, an important limitation for industrial applications, mostly when the primary acoustic ®eld is very high. Thus it seems that the optimization of the control sources positioning is an important factor, in order to minimize the power used. One of the objectives of this paper is to quantify the importance of the limited power for various parameters: frequency, ducts length, control sources positioning and error sensors plane. The ®rst section will present an experimental analysis of the distribution pressure ®eld in a semi-open duct. The eect of the source position and of the duct length on the interference pattern will be analyzed from an active control point of view. The second section will study an analysis of the acoustic power used by actuators in a multichannel system case. The eects of closed-end re¯ections, actuators position and error sensors position on the power used for the control sources will be analyzed. With this in mind, a positioning method has been developed and evaluated by experimental tests to minimize the power used by control actuators. 2. Longitudinal distribution of acoustic pressure ®eld In the case of the inside duct sound propagation, it is common knowledge that the stationary waves are established to form an interference pattern, following the re¯ection coecients and radiation impedance at the duct end [4±7]. Understanding this phenomenon of propagation is relatively simple for low frequencies when the plane mode (0,0) is the only one to be propagated in the duct. However, when several propagation modes occur, the inside duct pressure ®eld distribution becomes more complex because of the modes combination. In those cases, the pressure ®eld is not uniform, not only following the duct axis, but also in each duct section. Before analyzing the power actuators problem for the multichannel active control, it appears important to review the inside duct pressure ®eld longitudinal distribution for the higher order modes. 2.1. Experimental assembly To study the inside duct pressure ®eld longitudinal distribution, an experimental study has been realized with the assembly shown in Fig. 1. The pressure ®eld is generated in a PVC duct of 30 cm diameter and 2.05 m in length, with a closed and open end. A ®xed speaker on the duct wall is used to generate a pressure ®eld in the duct. The measurements were taken with the help of an acquisition system which uses a ®ve rotary microphones probe. A rod following the duct length axis is used to rotate the ®ve microphones, which are placed on a support in the duct radius axis. These ®ve microphones are distributed in uniform way, along the radius. A variable
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step electric motor is used to rotate the rod and to move the ®ve microphones, at every 10 . A multiplexer is installed to make the transition from one microphone to another. This assembly allows to choose the measurement plane according to the duct longitudinal axis. A white noise between 100 Hz and 2 kHz is generated by the speaker installed on the duct wall. The noise level recordings have been done using a program on a PC. This program records 801 lines of an autospectrum, with the HPIB port of an analyzer (BK2034). For the studied frequency range (100 Hz to 2 kHz), some higher order propagation modes are found. The cut-o frequency of the ®rst mode for the dimensions of the duct is 675 Hz. For the case of the frequencies higher than 675 Hz, the pressure ®eld in a section of the duct is not uniform. The measurements for each microphone allow to draw the noise level distribution of one frequency for a given duct section. For instance, Fig. 2a and b gives, respectively, the pressure ®eld distribution for a section at 800 and 1300 Hz. In Fig. 2a, it is easy to recognize the creation of a nodal line following the duct diameter, since the mode (0,0) and (0,1) are the only ones to be propagated. The combination of modes (0,0), (0,1) and (0,2) at 1300 Hz generates
Fig. 1. Measurement of acoustic pressure ®eld in a section: experimental assembly.
Fig. 2. (a) Distribution of noise levels in a section at 800 Hz, and (b) 1300 Hz.
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a complex pressure ®eld distribution that makes it dicult to identify the nodal lines. Since it is dicult to study the longitudinal distribution of the noise levels from a section to another, it seems preferable to use the measured mean level, in one section, as an indicator. 2.2. Measurements results 2.2.1. Overview Fig. 3a±d present examples of mean noise level distribution following the duct axis at 800, 1000, 1200 and 1400 Hz, respectively. The global mean level inside the duct varies signi®cantly with the frequency. These results show the interference pattern following the duct axis. However, these patterns become lower for high frequencies. Indeed, the dierence between maximum and minimum mean level goes from 14 to 4 dB between 800 and 1400 Hz. 2.2.2. Eect of the actuator position on the interference pattern The source position following the duct axis has an eect on the interference pattern identi®ed before. To illustrate this phenomenon, two positioning con®gurations for the noise source are presented (see Fig. 4). Because the interference patterns are well de®ned at low frequencies, a frequency of 800 Hz will be used to describe the eect of the source position. Fig. 5 shows the results of the two positions from Fig. 4. The noise source position following the duct axis has no eect on the interference pattern position and shape. On the other hand, the mean pressure level for the ®rst con®guration is approximately 5 dB higher than the one from the second con®guration. On primary analysis, we understand that this general diminution of the pressure ®eld results from the phase between the direct and re¯ected waves for the ®rst mode on the duct closed end (see Fig. 6). For a phase of approximately 180 , the progressive wave pressure is minimized, even though a phase of 0 or 360 increases that pressure. For higher frequencies, upper modes appear and the relative eect of these variations (associated to the plane mode) are reduced. Thus, at low frequencies, when a small number of modes are propagated, and the predominance of the plane mode continues to be important, we see that the source position plays a signi®cant role on the global mean level inside the duct. For the ®rst con®guration, the phase between emitted and re¯ected wave is ÿ14.5 . The second con®guration shows a phase of 140.5 . Thus, we can conclude that the con®guration #1 is a more ecient source position. 2.2.3. Eect of the duct length on the interference pattern The eect of the duct length on the interference pattern position and its shape has been evaluated with the help of a third con®guration shown on Fig. 7. The length of the duct has been reduced from 2.05 to 1.85 m in comparison to the one from con®gurations #1 and #2. The measurements of the mean levels are taken at the same place than the ones for the ®rst and the second con®guration. The closed end of the duct is used as a reference.
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Fig. 3. (a) Interference pattern versus duct axis at 800 Hz; (b) 1000 Hz; (c) 1200 Hz; and (d) 1400 Hz.
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Fig. 4. Con®gurations used for the primary source.
Fig. 5. Mean levels for case #1 and #2 used for primary source.
Fig. 6. Schematic explanation of the re¯ex wave's eect on the global noise level inside the duct.
Fig. 7. Con®guration #3.
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Fig. 8 shows the mean levels for the three con®gurations, for dierent measurement locations. From the results of the previous three subsections, the following conclusions have been made: a. The re¯ection of the plane mode from the duct closed end has a signi®cant eect on the pressure ®eld mean level inside the duct. b. The interference pattern position is not in¯uenced by the source position but only by the duct length. c. Finally, the amplitude of the interference pattern pressure ®eld inside the duct decreases at high frequencies. 2.3. Theoretical simulation The same kind of results have been obtained for the last three con®gurations with a theoretical model of pressure ®eld in a circular duct [4]. Fig. 9 shows results from the simulations for three con®gurations. Although the relation between the theoretical and experimental results are not perfect,1 the general tendencies are well respected. 3. Active control experimentation The results from the previous sections have shown that the noise source position governs the interference pattern amplitude, particularly at low frequencies This section will consider that problem, in the case of the multichannel active control. In the beginning, the plane mode re¯ection problems will be studied, with the help of control experimentation. The eect of the error sensors and the control sources plane position will be considered later.
Fig. 8. Mean levels for cases #1, #2 and #3 used for primary source. 1 The dierences between the theoretical and experimental results can be due, in part, to the interaction between the ¯uid and the assembly structure. Indeed, in comparison to the theoretical model with the rigid and perfect re¯ective wall assumptions, the assembly vibrates and transmits noise.
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3.1. Control problems in presence of re¯ections When the position of the control sources produces a phase of 180 between the emitted wave and re¯ected wave from the duct closed end, the power used by the control sources becomes more important. In order to explain this phenomenon, some control experimentations with and without re¯ections have been realized using the assembly shown in Fig. 10. To make these control experimentations, a feed-forward multichannel system has been used. This control algorithm running on a DSP C31 is linked to a PC board [8]. All the control speakers have the same position following the duct axis and are distributed in uniform way around the duct circumference. The active control experimentation will cover a frequency range from 1000 to 1350 Hz. For that interval, ®ve microphones and ®ve speakers are used. The number of channels used and the microphones disposition in the control plane are based on the methodology described in Ref. [4]. The control experimentations have been made using a pure tone with a frequency step of 10 Hz. The power used by the control sources has been estimated for each frequency. To study the eect of the re¯ections, two cases have been considered: the ®rst case with and the second one without absorbent material covering the duct closed end. For these control experimentations, the error microphones plane A has been used (see Fig. 10). The normalized electric power used by the control sources is calculated
Fig. 9. Mean levels for cases #1, #2 and #3 used for primary source (model).
Fig. 10. Assembly used for control experimentation.
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from rms voltage measures. These measurements are taken from each control source and are normalized by the primary source voltage: P
2rms sc
2rms sb
where: sb=noise source used to generate the primary ®eld sc=control sources. Fig. 11a and b give the normalized electric power used by each control source for the cases with and without absorbent material. For the normalized power higher than unity, the control sources require more acoustic energy than the primary source. Then, we have an overconsumption problem. These results show that the re¯ections from the duct closed end have an important eect on the power used by the control sources. Furthermore, the graph on Fig. 11 shows that the overconsumption problem is more important at low frequencies, and that it becomes blurred for higher frequencies because the plane mode interference becomes less and less important. 3.2. Eect of the sensor plane position Due to the interference pattern following the duct axis, the microphone plane can be on a maximum or a minimum of pressure (see Section 2.2). To check how the microphone plane's position on the interference pattern in¯uences the control sources consumption, a second microphone plane position (B) has been considered (see Fig. 10). The control experimentation has been done at 800 Hz. The graph on Fig. 12 shows the normalized power for the two microphone planes' positioning. The last graph does not show a lot of variations for the control sources used power between the two error sensors positioning. The mean noise reduction measurements at the open end with and without control show that the eciency is smaller when the control is applied on a low noise level zone (noise reductions: 31.1 versus 39.3 dB). This problem is related to the dynamic range of the input signal acquisition system. In a minimum pressure location, the microphones become more rapidly in measurement noise and, in this way, the control is less ecient. These results bring to the conclusion that the control sources used power reduction with the help of the error sensor plane positioning seem to be impossible. 3.3. Eect of the control sources' position The phase between the direct and re¯ected wave from the duct closed end seems to be the most in¯uential parameter for the control sources eciency. If the positioning of the control sources is in such a way that the phase is close to 0 , it seems possible to minimize the used power.
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Fig. 11. (a) Electrical power used versus frequency for a ®ve channel active control system in presence of re¯ections from the closed end. (b) Electrical power used versus frequency for a ®ve channel active control system without the presence of re¯ections from the closed end.
However, it is not practical, on the experimental assembly, to change the primary and control sources position. The method used for this paper consists in doing a study of the phase between the emitted and re¯ected wave from the duct closed end versus the frequencies for the con®guration showed in Fig. 10. Fig. 13 shows the phase versus the frequencies for two speakers planes and for the geometric conditions of Fig 10. To make easier the analysis of the frequencies range for which the sources positions are optimum, the graph of Fig. 14 has been built.
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Fig. 12. Electrical power used by each sources for sensor plane's position A and B.
Fig. 13. Phase between the primary source and the control sources versus frequency.
Fig. 14. Dierence of phase versus frequency.
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Fig. 15. Electrical power used by the control sources at 800 and 920 Hz.
This last graph shows the dierence between the absolute value of the plane mode phase for the noise source and the control sources. It is easy to judge which kind of source is more ecient. The positive zones show the frequencies where the control sources eciency is predominant, and the negative zones indicate that the noise source is more eciency. At 920 Hz, the control sources used power is smaller compared to 800 Hz. In fact, Fig. 14 shows a better eciency for the noise source at 800 Hz. To con®rm that the control sources used power is lower at 920 Hz, a control experimentation has been done. The used geometric con®guration is showed in Section 3.2. The results of the control experimentation are showed in Fig. 15. This graph shows the normalized power for 920 and 800 Hz (for microphone plane A, see Fig. 10). The noise reduction is practically the same at 920 and 800 Hz but, at 920 Hz, the control sources used power is lower. The maximum normalized power at 920 Hz is less than 0.20. For these two frequencies, two modes are being propagated and the pressure ®eld complexity is similar. 4. Conclusion For a multichannel control system, the control sources used power may become a real limitation problem. To analyze this problem, an assembly has been realized in laboratory. This assembly allows to analyze the pressure ®eld longitudinal distribution inside the duct. Many multichannel control experimentations for various frequencies, control sources positions and error sensors plane positions have been realized. The results of these experimentations show that the mean pressure ®eld longitudinal distribution has an important variation, and that the interference pattern position depends on the duct length and not on the noise source position. These variations are generated by the plane mode interference. This phenomenon becomes
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less important at high frequencies because the plane mode importance becomes smaller in comparison to the higher order propagation modes. The eciency of a noise source can be modulated by its position following the duct axis. It seems to be possible to optimize the control sources eciency if the plane mode phase between direct and re¯ected wave at the duct closed end is taken into consideration. This control sources optimization method has been checked in laboratory with a multichannel control system, and the obtained results have demonstrated the validity of this approach. Thus, this work allows to install a multichannel active control where the control sources acoustic power is an important constraint. References [1] Baumann DC, Grenier RA. Modal identi®cation approach to multi-modal cancellation. Proc. of Inter-noise 92. p. 341±8. [2] Erikson L. Higher order mode cancellation in ducts using active noise control. Proc. of Inter-noise 89. [3] L'EspeÂrance A, Bouchard M, Paillard B, Guigou C. US patent No. 5, 748, 750±May 5, 1998: Method and apparatus for active noise control of high order modes in ducts. [4] L'EspeÂrance A, Bouchard M, Paillard B, Guigou C, Boudreau A. Active noise control in large circular duct using an error sensors plane. Applied Acoustics 1999;57:357±74. [5] Levine H, Schwinger J. On the radiation of sound from an un¯anged circular pipe. Physical Review 1948;73(4):383±496. [6] Houiel F. ModeÂlisation du champ de pression et simulation de controÃle actif multicanal dans un conduit circulaire. MaõÃtrise es sciences appliqueÂes, speÂcialiteÂ: geÂnie meÂcanique; Groupe d'Acoustique et Vibrations de l'Universite de Sherbrooke, 1995. [7] Homicz GF, Lordi JA. A note on the radiative patterns of duct acoustics modes. J Sound Vib 1975;41(3):283±90. [8] Bouchard M. EÂtude d'algorithmes d'adaptation rapide pour controÃleurs actifs multi-variables. MaõÃtrise es sciences appliqueÂes, speÂcialiteÂ: geÂnie eÂlectrique; Groupe d'Acoustique et Vibrations de l'Universite de Sherbrooke, 1995.