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An electro-acoustics impedance error criterion and its application to active noise control
Applied Acoustics 65 (2004) 485–499 www.elsevier.com/locate/apacoust
An electro-acoustics impedance error criterion and its application to active noise control H. Hou
a,*
, J.H. Yang
b
a b
Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Automatic Control, Northwestern Polytechnical University, Xi’an 710072, PR China Received 20 January 2003; received in revised form 28 July 2003; accepted 4 November 2003
Abstract Characteristics of radiation impedance and its inducing variation of electrical impedance for a controllable source have been investigated. An impedance-based error criterion has been proposed and its application to active noise control is demonstrated through a coil driven loudspeaker. A general formula of radiation impedance is derived for two control strategies, according to the criterion of total acoustic power output. The radiation impedances of some commonly used sound sources are calculated. We discuss in detail the relation between variation of the input electrical impedance and radiation impedance for the two control strategies. An AC-bridge circuit is designed to measure the weak variation of electrical impedance resulted from radiation impedance. The input electrical impedance of a loudspeaker was measured and the experimental result is consistent with that of theoretical analysis. An impedance-based error criterion is proposed since the AC-bridge relative output is unique for a certain control strategy. The implementation of this criterion applied to an active control system is analyzed by simulations. An analogue control system is set up and experiments are carried out in a semi-anechoic chamber to verify the new control approach. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Electro-acoustic impedance; Error criterion; Active noise control
*
Corresponding author. E-mail address: [email protected] (H. Hou).
0003-682X/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2003.11.007
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1. Introduction Research works in active noise control (ANC) are generally divided into two respects, which are acoustic theory analysis and control system realization. Among these researches, selecting an effective control criterion is a key problem, which determines the physical limitation of sound reduction and the approach of control system implementation. Most existing error criterions in ANC, such as far-field sound pressure minimization or sound energy optimization, are all based on controllable parameters taken from sound field. Although there is a causal relation between source radiation and these error parameters, sound pressure or energy contains much more complicated factors from sound field circumstance. Therefore, the control of secondary source in such systems is performed indirectly. Researches show that ordinary ANC system works well only for relative simple primary source and sound field in free field or in enclosure. For a complex sound source or field, the control system is usually unstable due to sound feedback so that only a quiet area is achieved around the secondary source [1]. Recent researches have been involved in investigating different error criterions, including sound intensity, near-field sound pressure and membrane vibration velocity [2–4]. Many corresponding schemes of control system have appeared, which possess lots of advantages in complexity and stability. Impedance, defined as the ratio of force and its induced velocity, has been a wellaccepted concept. Radiation impedance (abbreviated as R-impedance) describes the sound radiation characters by self R-impedance and the sound reaction between primary and secondary sources by mutual R-impedance. Similarly, mechanical impedance (abbreviated as M-impedance) represents the dynamic characters of damp, stiffness and inertia of an electro-acoustic actuator. Using the electricity-mechanismacoustics analogy method, R-impedance, a part of M-impedance, may be transformed into electrical impedance (abbreviated as E-impedance). As we know, acquiring the transfer function of a sound source is an important problem for constructing a control system. Impedance, an equivalent transfer function, is a key parameter, which describes not only sound radiation ability, but also electrical driven character of a source actuator. So that, we may find a direct relation between sound radiation and source driven, which will reveal a novel error criterion based on impedance control. In sound field analysis of ANC, Elliott et al. [5] expressed sound power by acoustic impedance and derived the optimal source strength in impedance form. Compared with R-impedance, acoustic impedance describes the sound transmission character which is relative to sound medium other than sound source. Wang and Fuller [6] analyzed the mechanism of acoustic impedance variation in far field reduction of sound radiation from a vibrating plate when he compared two different secondary actuators of sound source and force source. Snyder and Hansen [7] demonstrated the essential of sound transmission control is impedance matching. Beyene and Burdisso [8] proposed the impedance matching technique was generally superior to the pressure-release method. Okada et al. [9,10] analyzed ANC in duct by means of impedance.
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Analysis and measurement of electro-acoustics parameters is a common interested problem, which is often carried out in a free sound field. Unfortunately, an analytical and experimental investigation of R-impedance and E-impedance of a sound source in ANC has not been found in the open literature. The objective of the work is to assess the feasibility of the impedance-based error criterion and its application to ANC. Firstly, a general formula of R-impedance is derived for two sound control strategies. Secondly, we analyzes the variation of E-impedance due to R-impedance change for a coil driven loudspeaker. Finally, an impedance criterion is proposed and the implementation of its application to a control system is discussed in much detail.
2. Radiation impedance in active noise control R-impedance is defined as the ratio of force and its induced velocity for an impulse source. In an interactional sound system of primary and secondary sources, their R-impedances are ( ZP ¼ FUPPP þ FUPSP ¼ ZPP þ UUPS ZPS ; ð1Þ ZS ¼ FUSSS þ FUSPS ¼ ZSS þ UUPS ZSP ; where F , U , Z are surface force, surface velocity and radiation impedance, respectively. Subscript ÔPÕ and ÔSÕ denote separately the primary and secondary source. R and X are real part and imaginary part of an impedance. In Eq. (1), ZPP ¼ RPP þ jXPP is self-impedance of primary source, which is determined by sourceÕs size and working frequency. ZSS ¼ RSS þ jXSS is self-impedance of secondary source. ZPS ¼ RPS þ jXPS is mutual impedance from secondary source to primary source, which is related to the distance between the two sources. ZSP ¼ RSP þ jXSP is mutual impedance from primary source to secondary source. Eq. (1) shows that total R-impedance rely not only on self-impedance and mutual impedance, but also on the ratio of velocity between the primary and secondary sources. Total sound power can be expressed by the R-impedance as following. WT ¼ 12Re½ZP jUP j2 þ 12Re½ZS jUS j2 :
ð2Þ
Optimal velocity of secondary source can be derived from taking the maximal sound power. It is US ¼
ðZPS þ ZSP Þ UP ; 2RSS
ð3Þ
where ‘‘*’’ denotes complex conjugate. 2.1. Global sound reduction If a strong mutual action exists between the primary and secondary sources, reciprocity comes into existence. It has
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ZPS ¼ ZSP :
ð4Þ
Substituting Eq. (4) into Eq. (3), one has the optimal secondary source strength US ¼
RSP UP : RSS
ð5Þ
In this strength, the optimal R-impedances of primary and secondary sources are 8 < ZP ¼ RPP RPS RSP þ j XPP XPS RSP ; RSS RSS ð6Þ : ZS ¼ j XSS XPS RSS : RSP We see from Eq. (6) that the R-impedance of secondary source is a pure imaginary number, which means the primary sound energy is converted by secondary source. This is the case of a globe sound reduction strategy, namely active sound restraining. 2.2. Local sound reduction When secondary source is far away from primary source or it is singledirectionality, only one-way effect from primary to secondary source exists so that the sound system is not reciprocity. Expressed in R-impedance, it has ZPS ¼ 0:
ð7Þ
Substituting Eq. (7) into Eq. (3), the optimal secondary source strength is given. US ¼
ZSP UP : 2RSS
ð8Þ
From Eq. (8) and Eq. (1), R-impedances of primary and secondary sources in this situation are ZP ¼ ZPP ; ð9Þ ZS ¼ RSS þ jXSS : Eq. (9) shows that the resistance of secondary source is a pure negative number that means the secondary source absorb primary sound energy and a quiet area appears around the secondary source. It is the case of local sound reduction strategy, namely active sound absorption.
3. Calculation examples 3.1. Monopole and monopole source system Both primary and secondary sources are assumed monopoles. The distance between them is d. R-impedances in active sound restraining strategy are [11]: ZP ¼ RPP ½1 sin c2 ðkdÞ þ jXPP 1 kaP sinkdkd coskdkd ; ð10Þ kd ; ZS ¼ jXSS 1 kaS cos sin kd
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Fig. 1. Amplitude of frequency-dependent R-impedance (non-dimensionalized by self-impedance): (a) primary monopole source, (b) secondary monopole source.
where k ¼ 2pf =c is wave number and S ¼ 4pa2 is the surface area of a monopole source. Fig. 1 gives the variation of relative R-impedance, ZP =ZPP and ZS =ZSS , with frequency f for active sound restraining. From curves we learn that resistance of secondary source is zero and resistance of primary source is smaller than its selfresistance when the two sources are close each other. There will be the best sound reduction when f ! 0 because the two sources construct a dipole source which has poor radiation efficiency. With f increases, R-impedance of secondary source keeps reactance and the resistance of primary source gradually approaches its self-resistance. In this situation, sound reduction is bad. 3.2. Monopole and dipole system For calculation simplicity, a monopole primary source is arranged on the axis of secondary dipole, as shown in Fig. 2. Only equivalent R-impedance can be obtained for directional source. Resistances of primary and secondary sources are [11]: 8 kd < RP ¼ RPP 1 3 coskdkd coskdkd sin ; k2 d 2 h i1 ð11Þ : RS ¼ RSS sin kd2 coskdkd sin kd2 : ðkdÞ ðkdÞ Fig. 3 shows the variation of relative R-resistance, RP =RPP and RS =RSS , with kd in ANC. For small kd, secondary resistance is negative that means the sound
Fig. 2. A primary monopole source is placed on the axis of a secondary dipole source.
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Fig. 3. Amplitude of frequency-dependent R-resistance (non-dimensionalized by self-resistance): (a) primary monopole source, (b) secondary dipole source.
absorption strategy will be achieved. But in the meanwhile, primary resistance is greater than its self-resistance – thus, the total power of primary and secondary source radiation keeps invariable. In this condition, the sound system corresponds to a tripole source which has an equal radiation efficiency as a monopole source when kd 1. Meanwhile, the resistance variations of primary and secondary sources imply the energy transmission between the two sources. In high frequency, there are lots of zero points in secondary resistance that is the typical character of sound restraining. 4. Analogy circuit and equivalent circuit for an electro-acoustic transducer In low frequency, a coil driven loudspeaker is regarded as a centralized vibration system. Its analogy circuit and equivalent circuit can be drawn by electricity– mechanism–acoustics analogy method, as shown in Fig. 4. In Fig. 4(a), Eg is signal voltage. Re and L are electrical resistance and inductance, respectively, which consist the static E-impedance Z0 ¼ Re þ jxL of the coil. Mm , Rm and Cm are mechanical equivalent mass, resistance and springiness respectively, 1 which form M-impedance Zm ¼ Rm þ jxMm þ jxC of the vibrating membrane. Bl is m efficiency coefficient from mechanism to electricity conversion. Rr , Xr is radiation resistance and reaction, where the vibration added mass Mr ¼ Xr =x. So that Zr ¼ Rr þ jXr stands for R-impedance of the loudspeaker transducer.
Fig. 4. (a) Analogy circuit of a coil driven loudspeaker, (b) electrical equivalent circuit of a coil driven loudspeaker.
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In order to analysis the electrical characteristics of the transducer. We draw the 2 electrical equivalent circuit as shown in Fig. 4(b), in which Rv ¼ ðBlÞ =ðRm þ Rr Þ, 2 2 Lv ¼ ðBlÞ Cm , Cv ¼ ðMm þ Mr Þ=ðBlÞ The total E-impedance of the transducer is Ze ¼ Z0 þ Zv ¼ Re þ jxL þ
1 : ð1=Rv þ jxCv þ 1=jxLv Þ
ð12Þ
In Eq. (12) Z0 is static E-impedance and Zv is dynamic E-impedance that comes from not only the membrane mechanical vibration but also sound radiation. Without sound radiation, total E-impedance of the transducer is determined by its mechanical characteristics. But with sound radiation, the R-impedance will change the dynamic impedance and hence make a variation of the total input E-impedance of the transducer.
5. The input E-impedance of a transducer in two control strategies In low frequency, the loudspeaker may be approximately regarded as monopole whose R-impedances in two control strategies are given in Eqs. (10) and (9). Substituting Eqs. (10) and (9) into Eq. (12), we can investigate the rule of the input E-impedances varying with R-impedance in two control strategies. Fig. 5 gives us the E-impedance characteristics of secondary source in sound restraining and sound absorption strategies, respectively. Compared with that before sound control, resonance position of E-impedance curve moves slightly and resonance amplitude increases a little because the zero R-resistance and pure R-reactance reduce mechanical damp and inertial mass of the vibration system, which results in variation of dynamic quality factor and resonance frequency. In sound absorption strategy, negative R-resistance decreases mechanical damp that leads to increase of dynamic quality factor so that position of peak value keeps immovable but the curve is more keen-edged.
Fig. 5. Amplitude of frequency-dependent E-impedance: off-control (solid line) and on-control (dished line), (a) active sound restraining, (b) active sound absorption.
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6. E-impedance measurement Experiments are carried out in a semi-anechoic chamber. A homemade AC-circuit bridge, as shown in Fig. 6, implements measurements of E-impedance for secondary source. The source we used is a coil driven speaker enclosed by a wooden cube whose size is 24 24 17 cm3 , the vibrating membrane of the speaker is 6 inch. Because the variation of E-impedance of the transducer due to R-impedance is a weak signal in the course of sound reduction, an AC-circuit bridge, which is well known in the field of electrical engineering, was designed to measure the tiny alteration of E-impedance before and after sound reduction. Z1 and Z2 are fixed bridge arms. Z3 represents the secondary source. Z4 is an adjustable E-impedance that ensures the bridge balance. Measurement steps: 1. To make the AC-circuit bridge balance and measure the E-impedance before sound control. 2. To adjust the strength of secondary source to achieve sound control strategies of active restraining and active absorption, respectively. 3. To make the AC-circuit bridge balance again and measure the E-impedance in two sound control strategies. The E-impedance measurements for secondary source are carried out in eight discrete frequencies from 120 to 400 Hz with 40 Hz as the step. Experimental results are shown in Fig. 7. In experiments, the primary and the secondary loudspeakers are arranged face to face. The distance between their vibration membranes is 4 cm for sound restraining strategy. In Fig. 7, both E-resistance and E-reactance in restraining strategy are smaller than that in off-control situation, which is consistent with analytical result. Although the absolute measured values of E-resistance and E-reactance are not equal to their responding computed values in each frequency, the trend of impedance
Fig. 6. (a) Experiment setup of E-impedance measurements, (b) AC-circuit bridge used in (a).
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Fig. 7. E-resistance: experimental result in (a), analytical result in (c) and E-reactance: experimental result in (b), analytical result in (d). Off control (* and ––) and on control ( and - - -) for active sound restraining.
changing with frequency is similar. The main error is the monopole source assumption and measurement error from the sensitivity of bridge circuit. Also in the experiments, we found the E-impedance variation of secondary source is quite small in the situation of sound absorption strategy because the distance between primary and secondary sources is rather far away.
7. Impedance criterion In Section 6, we measured the E-impedance in two strategies. In order to investigate the characteristic of E-impedance in the process of active sound reduction, we analyze the regulation of AC-circuit bridge parameters in the following. The bridgeÕs balance set before sound reduction will be broken owing to primary sound disturbance. The bridge output U_ cd resulted from tiny vary of R-impedance of secondary source will be U_ cd ¼
DZ3 DZ3 U_ S ¼ K U_ S ; Z Z3 3 ð1 þ Z1 =Z3 Þ Z1 =Z3
2
ð13Þ
where K ¼ ðZ1 =Z3 Þ=ð1 þ Z1 =Z3 Þ2 is a magnification coefficient which is dependent on bridge arms. U_ S is the voltage exerted on the bridge. Dot () in Eq. (13) is used for
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complex number. For a coil driven speaker, E-impedance can be expressed by its R-impedance. DZ3 , the variation of E-impedance of secondary source, is related to the variation of R-impedance DZR in process of sound reduction. One has DZ3 Zv DZR ¼ aDZR ; ¼ 2 Z3 ðBlÞ ð1 þ Z0 =Zv Þ
ð14Þ
2
where a ¼ Zv =ðBlÞ ð1 þ Z0 =Zv Þ is a proportion coefficient which is determined by electro-acoustics conversion characteristic of the bridge in balance situation before sound reduction. Other parameters are same as that illustrated in Section 6. Substituting Eq. (14) into Eq. (13) gives U_ cd ¼ KaDZR ; U_ S
ð15Þ
where K and a are constants for a given AC-bridge and secondary source structure, DZR is defined for a certain sound reduction strategy which we have analyzed in Section 2. The ratio of bridge output and its input, entirely determined by the variety of R-impedance of secondary source before and after control, is unique in two strategies, which implies a control law of impedance-based criterion in ANC. Primary source and secondary source are assumed both monopoles with same structure and electro-acoustics characteristics. The variation of R-impedance for secondary source can be derived as
Fig. 8. Bridge output: amplitude in (a), phase in (b) in different frequency for restraining (––) and absorption (- - -), for a certain frequency 300 Hz: bridge input in (c), outputs in (d) for restraining (––) and absorption (- - -).
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VP sinðkdÞ cosðkdÞ DZR ¼ ZS ZSS ¼ ZSS þ RSS þj ZSS kd kd VS sinðkdÞ cosðkdÞ VP ¼ RSS : þj kd kd VS
495
ð16Þ
From Section 2, we know the active sound restraining strategy will be achieved when the optimal strengths of primary and secondary sources satisfy VS =VP ¼ sinðkdÞ=kd. While, the sound absorption strategy is realized when VS =VP ¼ ½sinðkdÞ=ð2kdÞ þ j cosðkdÞ=ð2kdÞ. Substituting these results into Eq. (16), the impedance-based criterion for two control strategies will be ( h i U_ cd KaRSS 1 þ j cosðkdÞ ; sinðkdÞ ¼ ð17Þ U_ S 2KaRSS : Fig. 8 shows the relations between amplitude, phase of U_ cd =U_ S and frequency in two control strategies. The curves indicate the bridge output in sound restraining is greater than that in sound absorption because of the stronger coupling reaction between primary and secondary sources for the former case.
8. Ratio of bridges output to its input in the process of sound reduction Either sound restraining or sound absorption is realized when the strength ratio between primary and secondary source take their optimal value. Supposing the strength of secondary source is fixed, the rule of bridgeÕs output can be investigated by adjusting primary source in the process from the beginning of primary sourceÕs action until the situation of two control strategies are achieved. The optimal sourceÕs strength is usually expressed in surface velocity form. In order to transform it into the form of input driven voltage on the source, we will discuss the electro-acoustics transform relation below. The equivalent circuit of a loudspeaker in mechanical form is shown in Fig. 9. Variables in Fig. 9 have the same meanings as that in Fig. 4. In low frequency, jxL is 2 ignored and L=ðBlÞ is cut circuit. The surface velocity can be derived as
Vc ¼
Eg Bl=Re 2
ðBlÞ =Re þ Rm þ Rr þ jxðMm þ Mr Þ þ 1=jxCm
¼
Eg Bl ; Re ðRmt þ jXmt Þ
Fig. 9. Mechanical equivalent circuit of a coil driven loudspeaker.
ð18Þ
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where, Rmt ¼ ðBlÞ =Re þ Rm þ Rr , Xmt ¼ xðMm þ Mr Þ xC1 m , are M-resistance and M-reaction of an electro-acoustics transducer, respectively. For the case of primary and secondary sources getting same structure and parameters, the relation of ratio between primary and secondary sources in driven electrical voltage and surface velocity is Egs Vcs ¼b : Egp Vcp
ð19Þ
The proportion
sin kd cos kd V ZM þ ZS ZM þ ZSS þ RSS kd þ j kd VPS ; b¼ ¼ ZM þ ZP ZM þ ZPP þ RPP sinkdkd þ j coskdkd VVS P
ð20Þ
where subscript ÔSÕ and ÔPÕ denote R-impedance of secondary and primary source, respectively. ÔMÕ denotes M-impedance of the transducer. From Eqs. (19) and (20), one has E VS a EPS b ¼ ; VP a b EES P
ð21Þ
where a ¼ ZM þ ZPP , b ¼ RPP sinkdkd þ j coskdkd . For analysis simplicity, primary source is assumed with the same structure as secondary source, so that ZSS ¼ ZPP . Substituting Eq. (21) into Eq. (16), the variety of R-impedance expressed by driven voltage is obtained. DZR ¼
ab b2 EEPS a EEPS b
:
ð22Þ
In the other hand, the driven voltage of secondary source is a direct ratio to the port voltage of the AC bridge. There is ES Z3 þ DZ3 ¼ : U_ S Z3 þ DZ3 þ Z4
ð23aÞ
Since DZ3 Z3 , Eq. (23a) turns to ES ¼
1 U_ S : 1 þ Z4 =Z3
ð23bÞ
From Eqs. (22) and (23b), setting the port voltage of the AC bridge as U_ S , a tiny variety of R-impedance DZR will be found when the driven voltage of primary source is adjusted. Substituting DZR into Eq. (15), output voltage of the bridge can be obtained. Fig. 10 gives the detail of the relation between bridgeÕs output U_ cd and primary sourceÕs input EP . In the simulations, the port voltage of the bridge is U_ S ¼ 1. The primary and secondary sources are closely arranged that means stronger coupling exists between them so that the sound restraining strategy can be implemented. System works in 300 Hz frequency.
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Fig. 10. Bridge output: amplitude in (a) and (b), phase in (c) and (d) vs. primary input driven voltage: amplitude in (a) and (c), phase in (b) and (d).
In Fig. 10, output voltage of the bridge is dependent on both amplitude and phase of primary input driven voltage. From Fig. 10(a) and (b), we find that amplitude of U_ cd increases monotonously with the amplitude of primary input driven voltage (the phase difference of primary and secondary input driven voltages is p in this case), but changes periodically with the phase of primary input driven voltage (the amplitude of primary input driven voltage is 1 V in this case). Fig. 10(c) and (d) give phase variation of U_ cd against primary input driven voltage EP . Anyway, as aforementioned there is a fixed value of the bridge output U_ cd in two strategies. From this way we can control the source to be an optimal strength and achieve the best sound reduction.
9. Experimental measurement The experiment setup is shown in Fig. 6. All the measurements are carried out in a semi-anechoic chamber. For convenience, the bridgeÕs arm Z1 and Z2 are pure E-resistances. Work frequency is 300 Hz. Amplitude of bridgeÕs output voltage is measured by a micro-volt meter and its phase is observed by an oscillograph. In experiment, the primary and secondary sources are same configurations, which are 6 inch. caliber loudspeakers closed by a wooden cube. The two sources are very closed with 5 cm distance between their surface membranes. Setting a suitable driven voltage for secondary source and making surely the bridge balance at the beginning, then turning on the primary source and adjusting the amplitude and phase of the
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Fig. 11. (a) Amplitude of bridge output (measurement *, analysis ––) vs. primary driven voltage, (b) phase of bridge output (measurement *, analysis ––) vs. primary driven voltage.
driven voltage of primary source, recording the bridgeÕs output voltage, results are illustrated in Fig. 11. In Fig. 11(a), there is p phase difference between the bridgeÕs input and the primary sourceÕs driven voltage. In Fig. 11(b), the driven voltage of primary source is hold around 2 V. Fig. 11 shows that bridgeÕs output changes with the increase of primary driven voltage. The agreement between experimental measurement and analytical result supports the idea of impedance-based control approach. If both of amplitude and phase of primary source are adjusted, the sound restraining strategy will appear. We found a maximum 30 dB sound reduction in free field. In this situation, the bridgeÕs output is an optimal value which could be used for constructing a new ANC scheme. 10. Conclusions In this paper, a new error criterion based on Electro-acoustic impedance is presented, in which the secondary source acts as both actuator and sensor. An ANC system using an analogue circuit is constructed for sound restraining strategy. Taking a loudspeaker as a transducer, the tiny variation of its E-impedance resulted from R-impedance is measured and controlled by a designed AC bridge circuit. Experimental results demonstrate the feasibility of this novel criterion.
Acknowledgements Financial support from the National Natural Science Foundation of China (Grant 19904008) and the Excellent Young Teachers Program of MOE, China (No. 2093) is gratefully acknowledged. References [1] Nelson PA, Elliott SJ. Active control of sound. London: Academic Press; 1992. [2] Qiu X, Hansen CH, Li X. A comparison of near-field acoustic error sensing strategies for the active control of harmonic free field sound radiation. J Sound Vib 1998;215:81–103.
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[3] Kang SW, Kim YH. A comparison of sound energy and sound power based active noise control strategies. J Acoust Soc Am 1993;95:2932–3. [4] Zander AC, Hansen CH. A comparison of error sensor strategies for the active control of duct noise. J Acoust Soc Am 1993;94:841–8. [5] Elliott SJ, Joseph P, Nelson PA, Johnson ME. Power output minimization and power absorption in the active control of sound. J Acoust Soc Am 1991;90:2501–12. [6] Wang BT, Fuller CR. Near field pressure, intensity and wave-number distributions for active structural acoustic control of plate radiation: theoretical and analysis. J Acoust Soc Am 1992;92:1489–98. [7] Snyder SD, Hansen CH. Mechanisms of active noise control by vibration sources. J Sound Vib 1991;147:519–25. [8] Beyene S, Burdisso R. A new hybrid passive/active noise absorption system. J Acoust Soc Am 1997;101:1512–5. [9] Okda J, Usagawa T, Ebata M. Analysis of active control of noise in duct by means of impedance. J Acoust Soc Jpn 1987;8(2). [10] Okda J, Nishimura T, Nishimura Y, Usagawa T, Ebata M. Active noise control system using motional feedback loudspeaker. J Acoust Soc Jpn 1991;12(6). [11] Hou H, Yang JH, Li H, Sun JC. An impedance-based control system and its application in active noise control, Part I : theoretical analysis. J Northwestern Polytech Univ 2000;18:557–61 [in Chinese].
An electro-acoustics impedance error criterion and its application to active noise control H. Hou
a,*
, J.H. Yang
b
a b
Department of Applied Physics, Northwestern Polytechnical University, Xi’an 710072, PR China Department of Automatic Control, Northwestern Polytechnical University, Xi’an 710072, PR China Received 20 January 2003; received in revised form 28 July 2003; accepted 4 November 2003
Abstract Characteristics of radiation impedance and its inducing variation of electrical impedance for a controllable source have been investigated. An impedance-based error criterion has been proposed and its application to active noise control is demonstrated through a coil driven loudspeaker. A general formula of radiation impedance is derived for two control strategies, according to the criterion of total acoustic power output. The radiation impedances of some commonly used sound sources are calculated. We discuss in detail the relation between variation of the input electrical impedance and radiation impedance for the two control strategies. An AC-bridge circuit is designed to measure the weak variation of electrical impedance resulted from radiation impedance. The input electrical impedance of a loudspeaker was measured and the experimental result is consistent with that of theoretical analysis. An impedance-based error criterion is proposed since the AC-bridge relative output is unique for a certain control strategy. The implementation of this criterion applied to an active control system is analyzed by simulations. An analogue control system is set up and experiments are carried out in a semi-anechoic chamber to verify the new control approach. Ó 2003 Elsevier Ltd. All rights reserved. Keywords: Electro-acoustic impedance; Error criterion; Active noise control
*
Corresponding author. E-mail address: [email protected] (H. Hou).
0003-682X/$ - see front matter Ó 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.apacoust.2003.11.007
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1. Introduction Research works in active noise control (ANC) are generally divided into two respects, which are acoustic theory analysis and control system realization. Among these researches, selecting an effective control criterion is a key problem, which determines the physical limitation of sound reduction and the approach of control system implementation. Most existing error criterions in ANC, such as far-field sound pressure minimization or sound energy optimization, are all based on controllable parameters taken from sound field. Although there is a causal relation between source radiation and these error parameters, sound pressure or energy contains much more complicated factors from sound field circumstance. Therefore, the control of secondary source in such systems is performed indirectly. Researches show that ordinary ANC system works well only for relative simple primary source and sound field in free field or in enclosure. For a complex sound source or field, the control system is usually unstable due to sound feedback so that only a quiet area is achieved around the secondary source [1]. Recent researches have been involved in investigating different error criterions, including sound intensity, near-field sound pressure and membrane vibration velocity [2–4]. Many corresponding schemes of control system have appeared, which possess lots of advantages in complexity and stability. Impedance, defined as the ratio of force and its induced velocity, has been a wellaccepted concept. Radiation impedance (abbreviated as R-impedance) describes the sound radiation characters by self R-impedance and the sound reaction between primary and secondary sources by mutual R-impedance. Similarly, mechanical impedance (abbreviated as M-impedance) represents the dynamic characters of damp, stiffness and inertia of an electro-acoustic actuator. Using the electricity-mechanismacoustics analogy method, R-impedance, a part of M-impedance, may be transformed into electrical impedance (abbreviated as E-impedance). As we know, acquiring the transfer function of a sound source is an important problem for constructing a control system. Impedance, an equivalent transfer function, is a key parameter, which describes not only sound radiation ability, but also electrical driven character of a source actuator. So that, we may find a direct relation between sound radiation and source driven, which will reveal a novel error criterion based on impedance control. In sound field analysis of ANC, Elliott et al. [5] expressed sound power by acoustic impedance and derived the optimal source strength in impedance form. Compared with R-impedance, acoustic impedance describes the sound transmission character which is relative to sound medium other than sound source. Wang and Fuller [6] analyzed the mechanism of acoustic impedance variation in far field reduction of sound radiation from a vibrating plate when he compared two different secondary actuators of sound source and force source. Snyder and Hansen [7] demonstrated the essential of sound transmission control is impedance matching. Beyene and Burdisso [8] proposed the impedance matching technique was generally superior to the pressure-release method. Okada et al. [9,10] analyzed ANC in duct by means of impedance.
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Analysis and measurement of electro-acoustics parameters is a common interested problem, which is often carried out in a free sound field. Unfortunately, an analytical and experimental investigation of R-impedance and E-impedance of a sound source in ANC has not been found in the open literature. The objective of the work is to assess the feasibility of the impedance-based error criterion and its application to ANC. Firstly, a general formula of R-impedance is derived for two sound control strategies. Secondly, we analyzes the variation of E-impedance due to R-impedance change for a coil driven loudspeaker. Finally, an impedance criterion is proposed and the implementation of its application to a control system is discussed in much detail.
2. Radiation impedance in active noise control R-impedance is defined as the ratio of force and its induced velocity for an impulse source. In an interactional sound system of primary and secondary sources, their R-impedances are ( ZP ¼ FUPPP þ FUPSP ¼ ZPP þ UUPS ZPS ; ð1Þ ZS ¼ FUSSS þ FUSPS ¼ ZSS þ UUPS ZSP ; where F , U , Z are surface force, surface velocity and radiation impedance, respectively. Subscript ÔPÕ and ÔSÕ denote separately the primary and secondary source. R and X are real part and imaginary part of an impedance. In Eq. (1), ZPP ¼ RPP þ jXPP is self-impedance of primary source, which is determined by sourceÕs size and working frequency. ZSS ¼ RSS þ jXSS is self-impedance of secondary source. ZPS ¼ RPS þ jXPS is mutual impedance from secondary source to primary source, which is related to the distance between the two sources. ZSP ¼ RSP þ jXSP is mutual impedance from primary source to secondary source. Eq. (1) shows that total R-impedance rely not only on self-impedance and mutual impedance, but also on the ratio of velocity between the primary and secondary sources. Total sound power can be expressed by the R-impedance as following. WT ¼ 12Re½ZP jUP j2 þ 12Re½ZS jUS j2 :
ð2Þ
Optimal velocity of secondary source can be derived from taking the maximal sound power. It is US ¼
ðZPS þ ZSP Þ UP ; 2RSS
ð3Þ
where ‘‘*’’ denotes complex conjugate. 2.1. Global sound reduction If a strong mutual action exists between the primary and secondary sources, reciprocity comes into existence. It has
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ZPS ¼ ZSP :
ð4Þ
Substituting Eq. (4) into Eq. (3), one has the optimal secondary source strength US ¼
RSP UP : RSS
ð5Þ
In this strength, the optimal R-impedances of primary and secondary sources are 8 < ZP ¼ RPP RPS RSP þ j XPP XPS RSP ; RSS RSS ð6Þ : ZS ¼ j XSS XPS RSS : RSP We see from Eq. (6) that the R-impedance of secondary source is a pure imaginary number, which means the primary sound energy is converted by secondary source. This is the case of a globe sound reduction strategy, namely active sound restraining. 2.2. Local sound reduction When secondary source is far away from primary source or it is singledirectionality, only one-way effect from primary to secondary source exists so that the sound system is not reciprocity. Expressed in R-impedance, it has ZPS ¼ 0:
ð7Þ
Substituting Eq. (7) into Eq. (3), the optimal secondary source strength is given. US ¼
ZSP UP : 2RSS
ð8Þ
From Eq. (8) and Eq. (1), R-impedances of primary and secondary sources in this situation are ZP ¼ ZPP ; ð9Þ ZS ¼ RSS þ jXSS : Eq. (9) shows that the resistance of secondary source is a pure negative number that means the secondary source absorb primary sound energy and a quiet area appears around the secondary source. It is the case of local sound reduction strategy, namely active sound absorption.
3. Calculation examples 3.1. Monopole and monopole source system Both primary and secondary sources are assumed monopoles. The distance between them is d. R-impedances in active sound restraining strategy are [11]: ZP ¼ RPP ½1 sin c2 ðkdÞ þ jXPP 1 kaP sinkdkd coskdkd ; ð10Þ kd ; ZS ¼ jXSS 1 kaS cos sin kd
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Fig. 1. Amplitude of frequency-dependent R-impedance (non-dimensionalized by self-impedance): (a) primary monopole source, (b) secondary monopole source.
where k ¼ 2pf =c is wave number and S ¼ 4pa2 is the surface area of a monopole source. Fig. 1 gives the variation of relative R-impedance, ZP =ZPP and ZS =ZSS , with frequency f for active sound restraining. From curves we learn that resistance of secondary source is zero and resistance of primary source is smaller than its selfresistance when the two sources are close each other. There will be the best sound reduction when f ! 0 because the two sources construct a dipole source which has poor radiation efficiency. With f increases, R-impedance of secondary source keeps reactance and the resistance of primary source gradually approaches its self-resistance. In this situation, sound reduction is bad. 3.2. Monopole and dipole system For calculation simplicity, a monopole primary source is arranged on the axis of secondary dipole, as shown in Fig. 2. Only equivalent R-impedance can be obtained for directional source. Resistances of primary and secondary sources are [11]: 8 kd < RP ¼ RPP 1 3 coskdkd coskdkd sin ; k2 d 2 h i1 ð11Þ : RS ¼ RSS sin kd2 coskdkd sin kd2 : ðkdÞ ðkdÞ Fig. 3 shows the variation of relative R-resistance, RP =RPP and RS =RSS , with kd in ANC. For small kd, secondary resistance is negative that means the sound
Fig. 2. A primary monopole source is placed on the axis of a secondary dipole source.
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Fig. 3. Amplitude of frequency-dependent R-resistance (non-dimensionalized by self-resistance): (a) primary monopole source, (b) secondary dipole source.
absorption strategy will be achieved. But in the meanwhile, primary resistance is greater than its self-resistance – thus, the total power of primary and secondary source radiation keeps invariable. In this condition, the sound system corresponds to a tripole source which has an equal radiation efficiency as a monopole source when kd 1. Meanwhile, the resistance variations of primary and secondary sources imply the energy transmission between the two sources. In high frequency, there are lots of zero points in secondary resistance that is the typical character of sound restraining. 4. Analogy circuit and equivalent circuit for an electro-acoustic transducer In low frequency, a coil driven loudspeaker is regarded as a centralized vibration system. Its analogy circuit and equivalent circuit can be drawn by electricity– mechanism–acoustics analogy method, as shown in Fig. 4. In Fig. 4(a), Eg is signal voltage. Re and L are electrical resistance and inductance, respectively, which consist the static E-impedance Z0 ¼ Re þ jxL of the coil. Mm , Rm and Cm are mechanical equivalent mass, resistance and springiness respectively, 1 which form M-impedance Zm ¼ Rm þ jxMm þ jxC of the vibrating membrane. Bl is m efficiency coefficient from mechanism to electricity conversion. Rr , Xr is radiation resistance and reaction, where the vibration added mass Mr ¼ Xr =x. So that Zr ¼ Rr þ jXr stands for R-impedance of the loudspeaker transducer.
Fig. 4. (a) Analogy circuit of a coil driven loudspeaker, (b) electrical equivalent circuit of a coil driven loudspeaker.
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In order to analysis the electrical characteristics of the transducer. We draw the 2 electrical equivalent circuit as shown in Fig. 4(b), in which Rv ¼ ðBlÞ =ðRm þ Rr Þ, 2 2 Lv ¼ ðBlÞ Cm , Cv ¼ ðMm þ Mr Þ=ðBlÞ The total E-impedance of the transducer is Ze ¼ Z0 þ Zv ¼ Re þ jxL þ
1 : ð1=Rv þ jxCv þ 1=jxLv Þ
ð12Þ
In Eq. (12) Z0 is static E-impedance and Zv is dynamic E-impedance that comes from not only the membrane mechanical vibration but also sound radiation. Without sound radiation, total E-impedance of the transducer is determined by its mechanical characteristics. But with sound radiation, the R-impedance will change the dynamic impedance and hence make a variation of the total input E-impedance of the transducer.
5. The input E-impedance of a transducer in two control strategies In low frequency, the loudspeaker may be approximately regarded as monopole whose R-impedances in two control strategies are given in Eqs. (10) and (9). Substituting Eqs. (10) and (9) into Eq. (12), we can investigate the rule of the input E-impedances varying with R-impedance in two control strategies. Fig. 5 gives us the E-impedance characteristics of secondary source in sound restraining and sound absorption strategies, respectively. Compared with that before sound control, resonance position of E-impedance curve moves slightly and resonance amplitude increases a little because the zero R-resistance and pure R-reactance reduce mechanical damp and inertial mass of the vibration system, which results in variation of dynamic quality factor and resonance frequency. In sound absorption strategy, negative R-resistance decreases mechanical damp that leads to increase of dynamic quality factor so that position of peak value keeps immovable but the curve is more keen-edged.
Fig. 5. Amplitude of frequency-dependent E-impedance: off-control (solid line) and on-control (dished line), (a) active sound restraining, (b) active sound absorption.
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6. E-impedance measurement Experiments are carried out in a semi-anechoic chamber. A homemade AC-circuit bridge, as shown in Fig. 6, implements measurements of E-impedance for secondary source. The source we used is a coil driven speaker enclosed by a wooden cube whose size is 24 24 17 cm3 , the vibrating membrane of the speaker is 6 inch. Because the variation of E-impedance of the transducer due to R-impedance is a weak signal in the course of sound reduction, an AC-circuit bridge, which is well known in the field of electrical engineering, was designed to measure the tiny alteration of E-impedance before and after sound reduction. Z1 and Z2 are fixed bridge arms. Z3 represents the secondary source. Z4 is an adjustable E-impedance that ensures the bridge balance. Measurement steps: 1. To make the AC-circuit bridge balance and measure the E-impedance before sound control. 2. To adjust the strength of secondary source to achieve sound control strategies of active restraining and active absorption, respectively. 3. To make the AC-circuit bridge balance again and measure the E-impedance in two sound control strategies. The E-impedance measurements for secondary source are carried out in eight discrete frequencies from 120 to 400 Hz with 40 Hz as the step. Experimental results are shown in Fig. 7. In experiments, the primary and the secondary loudspeakers are arranged face to face. The distance between their vibration membranes is 4 cm for sound restraining strategy. In Fig. 7, both E-resistance and E-reactance in restraining strategy are smaller than that in off-control situation, which is consistent with analytical result. Although the absolute measured values of E-resistance and E-reactance are not equal to their responding computed values in each frequency, the trend of impedance
Fig. 6. (a) Experiment setup of E-impedance measurements, (b) AC-circuit bridge used in (a).
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Fig. 7. E-resistance: experimental result in (a), analytical result in (c) and E-reactance: experimental result in (b), analytical result in (d). Off control (* and ––) and on control ( and - - -) for active sound restraining.
changing with frequency is similar. The main error is the monopole source assumption and measurement error from the sensitivity of bridge circuit. Also in the experiments, we found the E-impedance variation of secondary source is quite small in the situation of sound absorption strategy because the distance between primary and secondary sources is rather far away.
7. Impedance criterion In Section 6, we measured the E-impedance in two strategies. In order to investigate the characteristic of E-impedance in the process of active sound reduction, we analyze the regulation of AC-circuit bridge parameters in the following. The bridgeÕs balance set before sound reduction will be broken owing to primary sound disturbance. The bridge output U_ cd resulted from tiny vary of R-impedance of secondary source will be U_ cd ¼
DZ3 DZ3 U_ S ¼ K U_ S ; Z Z3 3 ð1 þ Z1 =Z3 Þ Z1 =Z3
2
ð13Þ
where K ¼ ðZ1 =Z3 Þ=ð1 þ Z1 =Z3 Þ2 is a magnification coefficient which is dependent on bridge arms. U_ S is the voltage exerted on the bridge. Dot () in Eq. (13) is used for
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complex number. For a coil driven speaker, E-impedance can be expressed by its R-impedance. DZ3 , the variation of E-impedance of secondary source, is related to the variation of R-impedance DZR in process of sound reduction. One has DZ3 Zv DZR ¼ aDZR ; ¼ 2 Z3 ðBlÞ ð1 þ Z0 =Zv Þ
ð14Þ
2
where a ¼ Zv =ðBlÞ ð1 þ Z0 =Zv Þ is a proportion coefficient which is determined by electro-acoustics conversion characteristic of the bridge in balance situation before sound reduction. Other parameters are same as that illustrated in Section 6. Substituting Eq. (14) into Eq. (13) gives U_ cd ¼ KaDZR ; U_ S
ð15Þ
where K and a are constants for a given AC-bridge and secondary source structure, DZR is defined for a certain sound reduction strategy which we have analyzed in Section 2. The ratio of bridge output and its input, entirely determined by the variety of R-impedance of secondary source before and after control, is unique in two strategies, which implies a control law of impedance-based criterion in ANC. Primary source and secondary source are assumed both monopoles with same structure and electro-acoustics characteristics. The variation of R-impedance for secondary source can be derived as
Fig. 8. Bridge output: amplitude in (a), phase in (b) in different frequency for restraining (––) and absorption (- - -), for a certain frequency 300 Hz: bridge input in (c), outputs in (d) for restraining (––) and absorption (- - -).
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VP sinðkdÞ cosðkdÞ DZR ¼ ZS ZSS ¼ ZSS þ RSS þj ZSS kd kd VS sinðkdÞ cosðkdÞ VP ¼ RSS : þj kd kd VS
495
ð16Þ
From Section 2, we know the active sound restraining strategy will be achieved when the optimal strengths of primary and secondary sources satisfy VS =VP ¼ sinðkdÞ=kd. While, the sound absorption strategy is realized when VS =VP ¼ ½sinðkdÞ=ð2kdÞ þ j cosðkdÞ=ð2kdÞ. Substituting these results into Eq. (16), the impedance-based criterion for two control strategies will be ( h i U_ cd KaRSS 1 þ j cosðkdÞ ; sinðkdÞ ¼ ð17Þ U_ S 2KaRSS : Fig. 8 shows the relations between amplitude, phase of U_ cd =U_ S and frequency in two control strategies. The curves indicate the bridge output in sound restraining is greater than that in sound absorption because of the stronger coupling reaction between primary and secondary sources for the former case.
8. Ratio of bridges output to its input in the process of sound reduction Either sound restraining or sound absorption is realized when the strength ratio between primary and secondary source take their optimal value. Supposing the strength of secondary source is fixed, the rule of bridgeÕs output can be investigated by adjusting primary source in the process from the beginning of primary sourceÕs action until the situation of two control strategies are achieved. The optimal sourceÕs strength is usually expressed in surface velocity form. In order to transform it into the form of input driven voltage on the source, we will discuss the electro-acoustics transform relation below. The equivalent circuit of a loudspeaker in mechanical form is shown in Fig. 9. Variables in Fig. 9 have the same meanings as that in Fig. 4. In low frequency, jxL is 2 ignored and L=ðBlÞ is cut circuit. The surface velocity can be derived as
Vc ¼
Eg Bl=Re 2
ðBlÞ =Re þ Rm þ Rr þ jxðMm þ Mr Þ þ 1=jxCm
¼
Eg Bl ; Re ðRmt þ jXmt Þ
Fig. 9. Mechanical equivalent circuit of a coil driven loudspeaker.
ð18Þ
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where, Rmt ¼ ðBlÞ =Re þ Rm þ Rr , Xmt ¼ xðMm þ Mr Þ xC1 m , are M-resistance and M-reaction of an electro-acoustics transducer, respectively. For the case of primary and secondary sources getting same structure and parameters, the relation of ratio between primary and secondary sources in driven electrical voltage and surface velocity is Egs Vcs ¼b : Egp Vcp
ð19Þ
The proportion
sin kd cos kd V ZM þ ZS ZM þ ZSS þ RSS kd þ j kd VPS ; b¼ ¼ ZM þ ZP ZM þ ZPP þ RPP sinkdkd þ j coskdkd VVS P
ð20Þ
where subscript ÔSÕ and ÔPÕ denote R-impedance of secondary and primary source, respectively. ÔMÕ denotes M-impedance of the transducer. From Eqs. (19) and (20), one has E VS a EPS b ¼ ; VP a b EES P
ð21Þ
where a ¼ ZM þ ZPP , b ¼ RPP sinkdkd þ j coskdkd . For analysis simplicity, primary source is assumed with the same structure as secondary source, so that ZSS ¼ ZPP . Substituting Eq. (21) into Eq. (16), the variety of R-impedance expressed by driven voltage is obtained. DZR ¼
ab b2 EEPS a EEPS b
:
ð22Þ
In the other hand, the driven voltage of secondary source is a direct ratio to the port voltage of the AC bridge. There is ES Z3 þ DZ3 ¼ : U_ S Z3 þ DZ3 þ Z4
ð23aÞ
Since DZ3 Z3 , Eq. (23a) turns to ES ¼
1 U_ S : 1 þ Z4 =Z3
ð23bÞ
From Eqs. (22) and (23b), setting the port voltage of the AC bridge as U_ S , a tiny variety of R-impedance DZR will be found when the driven voltage of primary source is adjusted. Substituting DZR into Eq. (15), output voltage of the bridge can be obtained. Fig. 10 gives the detail of the relation between bridgeÕs output U_ cd and primary sourceÕs input EP . In the simulations, the port voltage of the bridge is U_ S ¼ 1. The primary and secondary sources are closely arranged that means stronger coupling exists between them so that the sound restraining strategy can be implemented. System works in 300 Hz frequency.
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Fig. 10. Bridge output: amplitude in (a) and (b), phase in (c) and (d) vs. primary input driven voltage: amplitude in (a) and (c), phase in (b) and (d).
In Fig. 10, output voltage of the bridge is dependent on both amplitude and phase of primary input driven voltage. From Fig. 10(a) and (b), we find that amplitude of U_ cd increases monotonously with the amplitude of primary input driven voltage (the phase difference of primary and secondary input driven voltages is p in this case), but changes periodically with the phase of primary input driven voltage (the amplitude of primary input driven voltage is 1 V in this case). Fig. 10(c) and (d) give phase variation of U_ cd against primary input driven voltage EP . Anyway, as aforementioned there is a fixed value of the bridge output U_ cd in two strategies. From this way we can control the source to be an optimal strength and achieve the best sound reduction.
9. Experimental measurement The experiment setup is shown in Fig. 6. All the measurements are carried out in a semi-anechoic chamber. For convenience, the bridgeÕs arm Z1 and Z2 are pure E-resistances. Work frequency is 300 Hz. Amplitude of bridgeÕs output voltage is measured by a micro-volt meter and its phase is observed by an oscillograph. In experiment, the primary and secondary sources are same configurations, which are 6 inch. caliber loudspeakers closed by a wooden cube. The two sources are very closed with 5 cm distance between their surface membranes. Setting a suitable driven voltage for secondary source and making surely the bridge balance at the beginning, then turning on the primary source and adjusting the amplitude and phase of the
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Fig. 11. (a) Amplitude of bridge output (measurement *, analysis ––) vs. primary driven voltage, (b) phase of bridge output (measurement *, analysis ––) vs. primary driven voltage.
driven voltage of primary source, recording the bridgeÕs output voltage, results are illustrated in Fig. 11. In Fig. 11(a), there is p phase difference between the bridgeÕs input and the primary sourceÕs driven voltage. In Fig. 11(b), the driven voltage of primary source is hold around 2 V. Fig. 11 shows that bridgeÕs output changes with the increase of primary driven voltage. The agreement between experimental measurement and analytical result supports the idea of impedance-based control approach. If both of amplitude and phase of primary source are adjusted, the sound restraining strategy will appear. We found a maximum 30 dB sound reduction in free field. In this situation, the bridgeÕs output is an optimal value which could be used for constructing a new ANC scheme. 10. Conclusions In this paper, a new error criterion based on Electro-acoustic impedance is presented, in which the secondary source acts as both actuator and sensor. An ANC system using an analogue circuit is constructed for sound restraining strategy. Taking a loudspeaker as a transducer, the tiny variation of its E-impedance resulted from R-impedance is measured and controlled by a designed AC bridge circuit. Experimental results demonstrate the feasibility of this novel criterion.
Acknowledgements Financial support from the National Natural Science Foundation of China (Grant 19904008) and the Excellent Young Teachers Program of MOE, China (No. 2093) is gratefully acknowledged. References [1] Nelson PA, Elliott SJ. Active control of sound. London: Academic Press; 1992. [2] Qiu X, Hansen CH, Li X. A comparison of near-field acoustic error sensing strategies for the active control of harmonic free field sound radiation. J Sound Vib 1998;215:81–103.
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