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A study of the analogue compensating monopole system for active attenuation of noise in a duct
Applied Acoustics 2g (1989) 187-202
A Study of the Analogue Compensating Monopole System for Active Attenuation of Noise in a Duct
Jiluo Zhou & Tielin Shi Department of Mechanical Engineering, Xi'an Jiaotong University, 710049 Xi'an, People's Republic of China (Received 14 November 1988; revised version received 2 June 1989; accepted 12 June 1989)
A BS TRA C T This paper describes theoretically the feedback effect of the secondary source on the primary source in a monopole system for active attenuation of noise in a duct, and gives a more accurate mathematical model Based on the model mentioned above, a new monopole system called the Analogue Compensating Monopole System, or ACM-System for short, is established. An analogue method .for compensating the non-ideal phase response of the system is suggested and discussed, and measuresJor improving the non-ideal amplitude response are proposed. In order to improve the working stability of the system, a directional sound detector is introduced, and the absolute stability condition of the A CM-System is given. Test results show that the method for compensating for the phase errors and improving the amplitude characteristics of the system is effective. Experimental data from the A CM-System in conditions of.flowing and quiet air in the duct are processed. Test results show that the ACM-System has a noise reduction ability superior to that of other monopole systems, whether in narrow or wider frequency bandwidths.
1 INTRODUCTION In m o d e r n noise c o n t r o l techniques, technicians are slowly taking m o r e interest in the active a t t e n u a t i o n system, because o f its small bulk, low cost and convenience in use. T h e basic principle o f the active a t t e n u a t i o n o f noise 187 Applied Acoustics 0003-682X/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
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is that the noise intensity will be reduced by the interference of two sound waves, or by inhibiting the radiation impedance of the sound source. In recent years the active attenuation system used for noise control in a duct has become more applicable due to its effectiveness at low frequencies, which is more difficult to realize with a general non-active noise control system. The main weakness of the active noise control system is that the effective attenuation bandwidth of noise is limited. Therefore, to widen the working bandwidth, and to improve the noise attenuation effect of the system are two important problems in studying active noise attenuation. The system demands not only effective attenuation, but low cost. In order to improve the attenuation effect of the system, the monopole system was developed several years ago along with the multipolar system. In multipolar systems two or more loudspeakers are used as the secondary source, so that the structure of the multipolar system is more complicated than that of the monopole system and therefore produces more storage errors. This makes the multipolar system work in a narrow bandwidth for the attenuation of noise and results in poor attenuation. Therefore, the multipolar system is limited in practical uses, and the research focus of the active noise attenuation system in recent years has turned again on the monopole system because of its simple structure and low cost. In studying the monopole system for active attenuation of noise in a duct, Eghtesadi and Leventhall 1 investigated the basic system and gave a primary mathematical model of the monopole system, but the attenuation effect is somewhat poor, and the working stability of the system is somewhat lower, owing to the influence of the non-ideal frequency response characteristics of the system components. Chaplin, 2"3 Warnaka, Tichy, 4'5 Martin, Roure 6 and Ross 7 improved the working ability of the system by using digital filter techniques and adaptive control techniques, by which both departures of the system components from ideal response, and the feedback from the secondary source to the detection microphone were compensated for, but the system is more expensive. In studying the response properties of the monopole system, this paper gives a more accurate mathematical model and proposes a new method for compensating the non-ideal response property of the monopole system. Further, a new monopole system called the Analogue Compensating Monopole System, or the ACM-System for short, is established. The ACMSystem can work in a wider frequency bandwidth of noise in a duct, and therefore a better attenuation effect can be achieved. On the basis of further theoretical analysis, a directional microphone is used for detecting the primary noise, with the result that the working stability of the system is greatly improved. Test results show that the theoretical analysis of the ACM-System is precise, and therefore, it can be applied satisfactorily in
A monopole system for active attenuation of noise & a duct
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practice. Another important point about the ACM-System is that it is simple and cheap in comparison with the Digital Compensating Monopole System.
2 SOME BASIC PRINCIPLES OF THE ACTIVE A T T E N U A T I O N OF NOISE IN A D U C T Figure 1 shows the sketch of a general monopole system for active attenuation of noise in a duct. When it is required that the noise downstream of the secondary source in the duct is to be fully attenuated, at least three conditions need to be satisfied. Firstly the sound frequency emitted by the Upstream Pr!mary
°0,5°
Downstream
Mic~_ 1 ,,too Pow..
I
0
Loo spo kor \J
DX
Fig. 1. The principle diagram of an ordinary monopole system for active attenuation of noise in a duct.
primary source should be lower than the corresponding cut-off frequency of the duct, in order that the sound wave propagates in the form of plane waves. Secondly the sound wave emitted by the secondary source should be delayed for time T, where z equals b/c (b is the distance from the primary source to the secondary source, c is the velocity of the sound wave), to allow for the travel time of the sound from the primary source to the secondary source, and the two sound waves emitted should be of equal amplitude and in opposite phase. That is, the amplitude ratio of the two sound waves should be exactly equal to 1, and the phase error for each of them should be equal to 0 in the frequency bandwidth. The third condition is that the sound detector of the system should be able to detect the primary noise only in one direction. This means that the directional coefficient a of the sound detector must be equal to 0. Consequently the sound wave emitted by the secondary source will return no more feedback to the detector, so the system will be prevented from self-oscillations. In practice, in the ordinary monopole system for active attenuation of noise in a duct, it is difficult to satisfy all three conditions mentioned above simultaneously. Firstly, the cut-off frequency of the duct is dependent on the
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Jiluo Zhou, Tielin Shi
cross-sectional area. The cut-off frequency of a duct with a square crosssection is: fm~x = c/2a
where a is the side-length of the square-sectional duct, in metres. Therefore, the size of the duct must be reduced in cross-section, in order to control the high frequency noise, but it can be used with difficulty on all occasions. The non-rigid duct wall and the damping effect existing during sound propagation in the duct can also influence the attenuation of the noise. Secondly, because of the errors that exist in each of the system components, it is impossible to make a system without an amplitude response error and phase response error, which influence the noise attenuation effect of the system. The better the required noise attenuation effect of the system, the smaller the system response errors permitted. The amplitude response error and the phase response error affect each other. That is, when the amplitude response error of the system is reduced to a certain value, the phase response error must be reduced accordingly, otherwise it is difficult to further improve the noise attenuation effect. Similarly, when the phase response error of the system is reduced to a certain value, the amplitude response error must be reduced accordingly. Therefore, control of the non-ideal property of the system components can result in good attenuation of noise. As far as the present condition is concerned, it is difficult to achieve complete attenuation of noise in a duct, even to a value of 30-40 dB, from a wide frequency bandwidth of noise by using the low cost analogue method. Thirdly, in the monopole system for active attenuation of noise in a duct, a one-directional sound detector can be used for eliminating the feedback effect of the sound which is emitted from the secondary source and acts on the sound detector. A more convenient method of eliminating the feedback effect of the secondary source is to modify the primary noise signal received by the sound detector and feed it into the secondary source. This enables the sound pressure downstream of the secondary source to approach zero. Assuming we take no account of the non-ideal property of system components and the damping effect of the sound propagating in the duct, the working and the transfer function block diagrams of the system shown in Fig. 2 will apply. The transfer function can be represented as: -
G(ioo)-
e-
i¢ot
1 + e -i2~°t
(1)
but this is not the transfer function required for complete noise attenuation. The transfer function for that purpose must be: G ( i ~ o ) = - e - i~,~
(2)
A monopole system for active attenuation o f noise in a duct
Primary I~: blc -I n°Ise'I'' ~Ac~ustic .... I feedbacJ
Primary
e"sx
(a)
191
-I
(b)
Fig. 2. (a) The working block diagram and (b) the transfer function block diagram of the ordinary monopole system.
However, on account of the non-ideal property of the system components, the system characteristics will still be affected by greater amplitude errors and phase errors, which greatly decrease the effect of the noise attenuation. Another disadvantage in the system is that the system will very easily become unstable and cause the system to whistle. Therefore, it is necessary to find other measures which will ensure that the system will work with high stability.
3 THE M A T H E M A T I C A L M O D E L AND ANALYSIS OF T H E ACM-SYSTEM The ACM-System proposed in this paper has two specific features. The first is that the system has considered, compensated and modified the effect of the non-ideal property existing in the system components. The other is that a one-directional sound detector is employed to eliminate the feedback effect from the secondary source, and this improves the system's working stability. Because of these two measures, attenuation of noise over wider frequency bandwidths will be possible. The system is now introduced below.
3.1 Analysis of the ACM-System transfer function It is known that all the components of the system, namely the microphone, the power amplifier, the loudspeaker, etc., have non-ideal response properties. Their response characteristics may be defined as:
HM(iog) is the ratio of the output voltage signals to the input sound pressure signals of the microphone; Hp(iOg) is the ratio of the output voltage signals to the input voltage signals of the power amplifier; HL(io9) is the ratio of the output sound pressure signals to the input voltage signals of the loudspeaker. Consequently, the transfer function block diagram of the ACM-System
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Primary noise
r. . . .
"1
....
I ~
F i g . 3.
The transfer function
r
I
~
block diagram
I~
feedback
of the ACM-System.
will be changed as shown in Fig. 3. The corresponding transfer function of the system is: HMHr, H L e i~'~ H°(i°9) = 1 + OtHMHeH e e - i2¢o.r -
(3)
From basic principles of the monopole system for active attenuation of noise in a duct, the frequency response of the system must be: H(iog) = - e - i ~ , , , as shown in eqn (2), if complete attenuation of noise is required. It is now actually feedback from the secondary source of the system and the non-ideal property of the components existing in the system which are affecting the attenuation result. The transfer function of the system will be Ho(ie)), shown in eqn (3). In order to eliminate the influence of these two factors, a compensating system with frequency response Hl(i~o ) is used to replace the element e -i'°~ framed with the dotted line as shown in Fig. 3. Then the transfer function of the compensating system will be: -- H M H p H L H 1 H°(i¢~) = 1 + ~ H M H v H L H 1 e-i,o~
(4)
To satisfy the requirement for complete noise attenuation in the duct, the transfer function Ho(i¢o) of the system represented in eqn (4) must be equal to - e -i~'* in the final form. Therefore, we have the transfer function: e - io,~ Hl(ico) = HA(1 -- ~ e-i2,o~)
(5)
where HA = H M H p H L . That is, the constant time delay z adopted in the original system must now be replaced by Hl(ico) represented in eqn (5). Assuming that HA = IHAle-~°'a~ and substituting into eqn (5), we have: e - ito(t - At)
Hl(io9) - IHAI(1 - ~ e-i2o~)
(6)
H~(ioo) can be obtained by a system whose working block diagram is shown in Fig. 4(a), and the transfer function block diagram is shown in Fig. 4(b). Obviously the compensating element H l ( k o ) of the system can eliminate the feedback effect of the secondary source in the upstream direction as well as compensate the non-ideal phase response of the system, but the structure
A monopole system for active attenuation of noise in a duct
193
~ e . _ S e- s( ' ~ - / Y I ; ) I ~ ~
p (a)
(b) Fig. 4.
(a) The working block diagram of the compensating segment H l(io ). (b) The transfer function block diagram of the ACM-System.
L
Power
amplifier White
L
LI
F
"~
noise
generator
(a) Sound level of : 11OdB g > x3
f----_
~.-
- 105 dB
> >
L 0
i°
~ 0
~
200
400 600 800 Frequency (Hz) (b)
95dB
I
1000
Fig. 5. (a) The block diagram for tests of HA(/to); (b) test results of the frequency amplitude response of H^(ito).
Jiluo Zhou, Tielin Shi
194
of the system is somewhat complicated. Experiments have proved that the transfer function of HA can be simplified and obtained by tests according to Fig. 5(a), and test results of [HAl used in this paper are shown in Fig. 5(b). From Fig. 5(b), it can be seen that the amplitude response of [HA[ approximates to a straight line, errors of which are within 2 dB in 1000 Hz bandwidth. Therefore, IHAI can be regarded as a constant A within a limited error. That is H A = A e -i~'a~. Substituting into eqn (6), we have: e - i,,,(, - a~)
Hl(i~) = A(1 - o~e-iz'~)
(7)
The time delay Ar in eqn (7) can be obtained by linear fitting of the phase response curve of H A shown in Fig. 6. The slope of the straight line is the
c
~.o
,.i-
100
I
I
200
300
I ~ l 400
Frequency Fig. 6.
500 (Hz)
600
Test results of the frequency phase response of HA(i~o).
constant time delay Az. The test result in this paper is Ar = 0"31 ms. Then, the transfer function of the compensating sub-system is: e - i~,~r- 0.000 31)
Hl(i~) = A(1 - ~ e-i2'°~)
(8)
The transfer function of Hl(i~o) in eqn (8) can be easily realized by using the integrated circuit. 3.2 The effect of the directional coefficient at of the sound detector on the system characteristics
In the discussions above, the feedback gain coefficient ~' in the transfer function Hl(ico ) shown in Fig. 4(b) has been considered as equal to the directional coefficient ~ of the sound detector, i.e. ~' = ~. In fact, the value of ~' is a constant, and the value of ~ fluctuates with the sound frequency.
A monopole system for active attenuation of noise in a duct
195
Both ~ and ~' are always unequal at different sound frequencies. When ct' q= ~t, eqn (5) becomes: e - i,~ = HA(1 -- 0t' e -i2~'~)
Hi(it°)
(9)
and the transfer function of the whole system is: -
e
-
i,o~
H ° ( i c ° ) = 1 - ~'(1 - cz/cx')e -i~°,
(10)
When 0(= ~, eqn (10) becomes: H o ( iCo ) = _ e - i~,~
This is just the transfer function required by the complete noise attenuation as shown in eqn (2). Otherwise, it can give both the amplitude and the phase errors for the system. The wider the frequency bandwidth is, the greater the fluctuation of~ and the greater the amplitude and phase errors of the system will be. Figure 7 gives the amplitude and the phase errors of the system, when ~' and ~ have different values. The greater the errors of the system, the more Cl{,° - ClL
:
40"1,
OL
CL = 0 . 5
~' ~o 3
•
5
n
I
200
e
I
400 Frequency (a)
~
~-2
800
600 800 Frequency (Hz)
-1
=
600 (Hz)
i
(b)
Fig. 7.
The phase errors and amplitude errors of the ACM-System under different sound frequencies and different values of (ct'-ct)/a, during ct = 0'5.
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Jiluo Zhou, Tielin Shi
difficult it is to improve the noise attenuation effect of the system in a wide bandwidth. There are three methods of limiting the difference of values between ~' and c~.The first is to select a sound detector with a constant directional coefficient ~. The second is to select an omnidirectional sound detector, whose directional coefficient ~ = 0 , but this is not practicable because of the instability of the system under working conditions. The third is to select a sound detector whose directional coefficient ~ = 1, i.e. the sound detector detects the noise signals only in one direction, and such a detector can be composed of two omnidirectional sound detectors.
3.3 Stability analysis of the ACM-System It is known that the stability problem arises in all closed-loop systems. The necessary and sufficient conditions for self-oscillations of the system are:
[Gl(ico G2(i~o)l > 1 ~ G I ( i ~ ) . G2(i~ ) = 2mz,
forn = 1,2,3 ....
(11)
where Gl(i~ ) is the transfer function of the forward feed channel of the system G2(i~o) is the transfer function of the feedback loop of the system. If one of the conditions represented in e q n ( l l ) is not met, selfoscillations will never arise again. From Fig. 4 we can find that the transfer function of the forward feed channel of the ACM-System is: --IHAle -/°'~
Gl(i°9) = 1 - ~'e -i2c°*
(12)
and the transfer function of the feedback loop of the: ACM-System is:
G2(i~o) = ~ e-i'°~
(13)
Then, the amplitude of the product of above transfer functions Gl(ioJ ) and G2(ico ) is:
IGl(i6o). G2(itn)l =
~IHAI x/1 - 2c( cos 2coz + ~,2
(14)
From eqn (14) we can obtain an absolute stable working condition of the ACM-System as:
~IHAI i x/1 - 2a' cos 2o~z + ~,2
< 1
(15)
A monopole system]br active attenuation of noise in a duct
197
Under the limited condition of cos 2o9r = 1, we get: sinAI
- - < 1 -- ~'
1
(16)
For a practical system for the active attenuation of noise in a duct, the feedback gain coefficient ~' is usually constant and it is c o m m o n to have IHAI~ 1. Substituting these values of ~' and IHAIinto eqn (16), the stability condition of the ACM-System can be obtained as: < 1 - ~'
(17)
With the condition ~ = ~', we get a < 0.5. This condition can be easily satisfied by the ACM-System.
4 TEST R E S U L T S A N D A N A L Y S I S O F T H E A C M - S Y S T E M The experimental set used for testing the working capacity of the system is illustrated in Fig. 8. The sound detector Mic 1 employed is sensitive only to the primary noise and not to the secondary noise or the reflections from the termination of the duct; the sound detector Mic 2 is applied to measure the attenuation results of the system. As to the working process of the whole system, it is easily understood from the block diagram shown in Fig. 8, and it is unnecessary to go into details. Three attenuation tests have been carried out for the ACM-System: the first is the pure tone noise attenuation test; the second is a random noise attenuation test; and the third test is processed with air flowing in the duct. These three tests are described in detail respectively as follows. / 4.1 Pure tone noise attenuation tests o f the A C M - S y s t e m
As we know noise in reality is always of a continuous spectrum. Such a noise spectrum can be regarded as a pile of several pure tone noise spectrums. The Primary
Power
~i
Mic1
MiCci
Second/~ry source
amplifier Sounder box
Signal
generator
Fig. 8,
~ J [ I
~ ~
~
Power amplifier
Spectrum I ana y s e r
I
The sketched experimental set diagram of the ACM-System for tests of its attenuation capacity.
Jiluo Zhou, Tielin Shi
198
c:
.o ¢)
"o
o
1oo
200
Frequency (Hz) Fig. 9.
Test results of the ACM-System for attenuation of a 100 Hz pure tone noise in a duct.
elimination of frequency peaks in the noise spectrum can effectively reduce the sound level of the noise. Figure 9 shows the test result for the attenuation of a 100 Hz pure tone noise in a duct. Curve a is the primary noise spectrum of the system taken before the secondary source is turned on; curve b is the noise spectrum after the secondary source is turned on during the test. (An attenuation test with acoustic material wedges placed just before the termination of a duct has also been processed. The test result shows a similar curve b to that shown in Fig. 9. It means that the much lower reflections from the termination has practically no influence on the test result.) Table 1 shows the test results for the attenuation of pure tone noise with different frequencies. The test results demonstrate that the ACM-System gives good attenuation for the noise in the duct. From Table 1 it can be seen that with increasing noise frequency the attenuation will be decreased. There are mainly two reasons for this: one is due to vibrations of the duct wall, the other is due to the error Ab of the installed distance b' and the theoretical distance b from the detector to the secondary source. The distance error Ab influences the attenuation more at high frequencies than low frequencies. Improvement of the stiffness of the duct wall and accurate control of the distance b from the detector to the TABLE 1 Pure Tone Noise Attenuation Results of the ACM-System During Test in Duct
Sound frequency, Hz Attenuated sound level AL, dB
I00 66
200 45
250 42
300 40
400 38
500 25
800 18
1 000 12
A monopole system for active attenuation of noise in a duct
199
secondary source greatly increases the attenuation of pure tone noise and reduces the noise peaks. 4.2 Random noise attenuation tests of the ACM-System As mentioned above, the main factor which affects the attenuation result is the phase and amplitude error caused by the non-ideal response property of the system components. The linear error can be decreased by the compensating transfer function H~(io~)of the ACM-System. And a specially designed closed type sounder box can be used to improve the radiation capacity of the loudspeaker at low frequencies, and this can result in decreasing the amplitude error. Tests have been performed with 100, 250, 500 and 1000Hz bandwidths respectively. Test results are shown in Fig. 10, and the corresponding
0 I
.__> 2 0 0
I
250
300
I
700
I
750
800
4; > 4;
0
%o -go
55o 6;o 6;o %o %o
> C 0 U~
100
I
I
I
I
I
200
300
400
500
600
Frequency Fig. 10.
0 (Hz)
I
I
I
200
400
600
/
800
I
1000
Test results of the ACM-System for attenuation of the random noise in a duct with different bandwidths.
Jiluo Zhou, Tielin Shi
200
attenuated values of noise are on average 30, 20, 15 and 12 dB respectively. The test results demonstrate that the ACM-System can give a good attenuation on noise of wide as well as narrow bandwidth, although of course, the narrow bandwidth gives better results. Therefore, the ACM-System is superior to any other general monopole system. 4.3 Random noise attenuation tests of the ACM-System with air flowing in a duct When the ACM-System is used with air flowing in a duct, the flowing air will affect the attenuation in the duct; it will influence the sound propagation, and noise detecting accuracy. The former can be neglected when the air flow velocity is low. When the air flow velocity is comparatively high, the effect can be eliminated by adjusting the time delay parameter z in the
"2 O [
> 100 '5
I
I
250
300
I
150
200
200
400
400
II1 "D
>
200
300
I 500
1 600
0
100
Fig. 11.
I
l
I
i
200
300
400
500
i
600 Frequency
I
0 (Hz)
i
200 400
i
I
I
600
800
1000
Test results of the ACM-System for attenuation of the random noise under conditions ofair flow in the duct with different bandwidths.
A monopole system for active attenuation of noise in a duct
201
compensating transfer function Hx(io9). The latter effect can be decreased by using a windshield or a sound wave travelling tube. Test results of the ACM-System obtained under conditions of air flow in the duct are shown in Fig. 11. The air flow velocity in the duct is 20m/s during the test. The sound detector is protected by a windshield in order to improve the testing accuracy. Test results demonstrate that the ACMSystem gives good noise attenuation under conditions of air flow in a duct. The attenuation of the ACM-System, especially for noise of low frequency, approaches the effect of noise in quiet air. From the test results mentioned above, it can be seen that the ACMSystem has the capacity to control the noise in a duct under different working conditions. 5 CONCLUSIONS (1) In modern active attenuation systems for noise in a duct, the monopole system is now under rapid development because of its superior characteristics due to its compact structure, low cost and ease of use. The research focus of the monopole system is still overcoming its weak points and improving the capacity of the system. (2) The amplitude and phase response errors resulting from the non-ideal properties of the system components are the main factors which affect the noise attenuation results of the monopole system. The amplitude response errors of the system, which are mainly produced by the loudspeaker, can be decreased many times by a special acoustic design for improving its radiation capacity. The phase response errors of the system can be corrected by designing a compensating segment Hl(iog) in the system. (3) Through a careful study of the compensating segment Hl(io9 ), it is shown that a rational construction of the sound detector, the power amplifier and the loudspeaker can make the response characteristics of the s y s t e m H A = H M H p H L = A e-i,,a~, where A is constant. This means that the detector, the power amplifier and the loudspeaker are all unnecessary for acquiring a better response. (4) A directional detector introduced in the monopole system can improve the working stability of the system, lfthe directional coefficient a of the sound detector and the feedback gain coefficient a' in the ACM-System are satisfied by eqn (17), i.e. a < 1 - a ' , the system will be fully stable under working conditions. (5) The ACM-System established according to the theory described above can have a better attenuation of noise than that of the other ordinary monopole system. It is also as good as the multipolar active attenuation system.
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Jiluo Zhou, Tielin Shi REFERENCES
1. Eghtesadi, Kh. & Leventhall, H. G., Active attenuation of noise--The monopole system. J. Acoust. Soc. of Am. 71(3) (1982) 608-11. 2. Chaplin, B., The cancellation of repetitive noise and vibration. In Proceedings of Inter-noise 80, Institute of Noise Control Engineering, USA, 1980, pp. 699-702. 3. Chaplin, B., An adaptive wave-form generator for cancelling engine exhaust noise. Report for the Institute of Electronic and Radio Engineers, UK, 1980. 4. Warnaka, G. E. & Tichy, J., Acoustic mixing in active attenuators. In Proceedings oflnter-noise 80, Institute of Noise Control Engineering, USA, 1980, pp. 683-8. 5. Tichy, J. & Warnaka, G. E., Effect of evanescent waves of the active attenuation of sound in ducts. In Proceedings of Inter-noise 83, Institute of Acoustics, UK, 1983, pp. 435-8. 6. Martin, V. & Roure, A., Control of sources for active sound propagation in a duct. In Proceedings of Inter-noise 83, Institute of Acoustics, UK, 1983, pp. 431-4. 7. Ross, C. F., An adaptive digital filter for broadband active sound control, J. Sound Vibr., 80(3) (1982) 381-8.
A Study of the Analogue Compensating Monopole System for Active Attenuation of Noise in a Duct
Jiluo Zhou & Tielin Shi Department of Mechanical Engineering, Xi'an Jiaotong University, 710049 Xi'an, People's Republic of China (Received 14 November 1988; revised version received 2 June 1989; accepted 12 June 1989)
A BS TRA C T This paper describes theoretically the feedback effect of the secondary source on the primary source in a monopole system for active attenuation of noise in a duct, and gives a more accurate mathematical model Based on the model mentioned above, a new monopole system called the Analogue Compensating Monopole System, or ACM-System for short, is established. An analogue method .for compensating the non-ideal phase response of the system is suggested and discussed, and measuresJor improving the non-ideal amplitude response are proposed. In order to improve the working stability of the system, a directional sound detector is introduced, and the absolute stability condition of the A CM-System is given. Test results show that the method for compensating for the phase errors and improving the amplitude characteristics of the system is effective. Experimental data from the A CM-System in conditions of.flowing and quiet air in the duct are processed. Test results show that the ACM-System has a noise reduction ability superior to that of other monopole systems, whether in narrow or wider frequency bandwidths.
1 INTRODUCTION In m o d e r n noise c o n t r o l techniques, technicians are slowly taking m o r e interest in the active a t t e n u a t i o n system, because o f its small bulk, low cost and convenience in use. T h e basic principle o f the active a t t e n u a t i o n o f noise 187 Applied Acoustics 0003-682X/89/$03"50 © 1989 Elsevier Science Publishers Ltd, England. Printed in Great Britain
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Jiluo Zhou, Tielin Shi
is that the noise intensity will be reduced by the interference of two sound waves, or by inhibiting the radiation impedance of the sound source. In recent years the active attenuation system used for noise control in a duct has become more applicable due to its effectiveness at low frequencies, which is more difficult to realize with a general non-active noise control system. The main weakness of the active noise control system is that the effective attenuation bandwidth of noise is limited. Therefore, to widen the working bandwidth, and to improve the noise attenuation effect of the system are two important problems in studying active noise attenuation. The system demands not only effective attenuation, but low cost. In order to improve the attenuation effect of the system, the monopole system was developed several years ago along with the multipolar system. In multipolar systems two or more loudspeakers are used as the secondary source, so that the structure of the multipolar system is more complicated than that of the monopole system and therefore produces more storage errors. This makes the multipolar system work in a narrow bandwidth for the attenuation of noise and results in poor attenuation. Therefore, the multipolar system is limited in practical uses, and the research focus of the active noise attenuation system in recent years has turned again on the monopole system because of its simple structure and low cost. In studying the monopole system for active attenuation of noise in a duct, Eghtesadi and Leventhall 1 investigated the basic system and gave a primary mathematical model of the monopole system, but the attenuation effect is somewhat poor, and the working stability of the system is somewhat lower, owing to the influence of the non-ideal frequency response characteristics of the system components. Chaplin, 2"3 Warnaka, Tichy, 4'5 Martin, Roure 6 and Ross 7 improved the working ability of the system by using digital filter techniques and adaptive control techniques, by which both departures of the system components from ideal response, and the feedback from the secondary source to the detection microphone were compensated for, but the system is more expensive. In studying the response properties of the monopole system, this paper gives a more accurate mathematical model and proposes a new method for compensating the non-ideal response property of the monopole system. Further, a new monopole system called the Analogue Compensating Monopole System, or the ACM-System for short, is established. The ACMSystem can work in a wider frequency bandwidth of noise in a duct, and therefore a better attenuation effect can be achieved. On the basis of further theoretical analysis, a directional microphone is used for detecting the primary noise, with the result that the working stability of the system is greatly improved. Test results show that the theoretical analysis of the ACM-System is precise, and therefore, it can be applied satisfactorily in
A monopole system for active attenuation of noise & a duct
189
practice. Another important point about the ACM-System is that it is simple and cheap in comparison with the Digital Compensating Monopole System.
2 SOME BASIC PRINCIPLES OF THE ACTIVE A T T E N U A T I O N OF NOISE IN A D U C T Figure 1 shows the sketch of a general monopole system for active attenuation of noise in a duct. When it is required that the noise downstream of the secondary source in the duct is to be fully attenuated, at least three conditions need to be satisfied. Firstly the sound frequency emitted by the Upstream Pr!mary
°0,5°
Downstream
Mic~_ 1 ,,too Pow..
I
0
Loo spo kor \J
DX
Fig. 1. The principle diagram of an ordinary monopole system for active attenuation of noise in a duct.
primary source should be lower than the corresponding cut-off frequency of the duct, in order that the sound wave propagates in the form of plane waves. Secondly the sound wave emitted by the secondary source should be delayed for time T, where z equals b/c (b is the distance from the primary source to the secondary source, c is the velocity of the sound wave), to allow for the travel time of the sound from the primary source to the secondary source, and the two sound waves emitted should be of equal amplitude and in opposite phase. That is, the amplitude ratio of the two sound waves should be exactly equal to 1, and the phase error for each of them should be equal to 0 in the frequency bandwidth. The third condition is that the sound detector of the system should be able to detect the primary noise only in one direction. This means that the directional coefficient a of the sound detector must be equal to 0. Consequently the sound wave emitted by the secondary source will return no more feedback to the detector, so the system will be prevented from self-oscillations. In practice, in the ordinary monopole system for active attenuation of noise in a duct, it is difficult to satisfy all three conditions mentioned above simultaneously. Firstly, the cut-off frequency of the duct is dependent on the
190
Jiluo Zhou, Tielin Shi
cross-sectional area. The cut-off frequency of a duct with a square crosssection is: fm~x = c/2a
where a is the side-length of the square-sectional duct, in metres. Therefore, the size of the duct must be reduced in cross-section, in order to control the high frequency noise, but it can be used with difficulty on all occasions. The non-rigid duct wall and the damping effect existing during sound propagation in the duct can also influence the attenuation of the noise. Secondly, because of the errors that exist in each of the system components, it is impossible to make a system without an amplitude response error and phase response error, which influence the noise attenuation effect of the system. The better the required noise attenuation effect of the system, the smaller the system response errors permitted. The amplitude response error and the phase response error affect each other. That is, when the amplitude response error of the system is reduced to a certain value, the phase response error must be reduced accordingly, otherwise it is difficult to further improve the noise attenuation effect. Similarly, when the phase response error of the system is reduced to a certain value, the amplitude response error must be reduced accordingly. Therefore, control of the non-ideal property of the system components can result in good attenuation of noise. As far as the present condition is concerned, it is difficult to achieve complete attenuation of noise in a duct, even to a value of 30-40 dB, from a wide frequency bandwidth of noise by using the low cost analogue method. Thirdly, in the monopole system for active attenuation of noise in a duct, a one-directional sound detector can be used for eliminating the feedback effect of the sound which is emitted from the secondary source and acts on the sound detector. A more convenient method of eliminating the feedback effect of the secondary source is to modify the primary noise signal received by the sound detector and feed it into the secondary source. This enables the sound pressure downstream of the secondary source to approach zero. Assuming we take no account of the non-ideal property of system components and the damping effect of the sound propagating in the duct, the working and the transfer function block diagrams of the system shown in Fig. 2 will apply. The transfer function can be represented as: -
G(ioo)-
e-
i¢ot
1 + e -i2~°t
(1)
but this is not the transfer function required for complete noise attenuation. The transfer function for that purpose must be: G ( i ~ o ) = - e - i~,~
(2)
A monopole system for active attenuation o f noise in a duct
Primary I~: blc -I n°Ise'I'' ~Ac~ustic .... I feedbacJ
Primary
e"sx
(a)
191
-I
(b)
Fig. 2. (a) The working block diagram and (b) the transfer function block diagram of the ordinary monopole system.
However, on account of the non-ideal property of the system components, the system characteristics will still be affected by greater amplitude errors and phase errors, which greatly decrease the effect of the noise attenuation. Another disadvantage in the system is that the system will very easily become unstable and cause the system to whistle. Therefore, it is necessary to find other measures which will ensure that the system will work with high stability.
3 THE M A T H E M A T I C A L M O D E L AND ANALYSIS OF T H E ACM-SYSTEM The ACM-System proposed in this paper has two specific features. The first is that the system has considered, compensated and modified the effect of the non-ideal property existing in the system components. The other is that a one-directional sound detector is employed to eliminate the feedback effect from the secondary source, and this improves the system's working stability. Because of these two measures, attenuation of noise over wider frequency bandwidths will be possible. The system is now introduced below.
3.1 Analysis of the ACM-System transfer function It is known that all the components of the system, namely the microphone, the power amplifier, the loudspeaker, etc., have non-ideal response properties. Their response characteristics may be defined as:
HM(iog) is the ratio of the output voltage signals to the input sound pressure signals of the microphone; Hp(iOg) is the ratio of the output voltage signals to the input voltage signals of the power amplifier; HL(io9) is the ratio of the output sound pressure signals to the input voltage signals of the loudspeaker. Consequently, the transfer function block diagram of the ACM-System
Jiluo Zhou, Tielin Shi
192
Primary noise
r. . . .
"1
....
I ~
F i g . 3.
The transfer function
r
I
~
block diagram
I~
feedback
of the ACM-System.
will be changed as shown in Fig. 3. The corresponding transfer function of the system is: HMHr, H L e i~'~ H°(i°9) = 1 + OtHMHeH e e - i2¢o.r -
(3)
From basic principles of the monopole system for active attenuation of noise in a duct, the frequency response of the system must be: H(iog) = - e - i ~ , , , as shown in eqn (2), if complete attenuation of noise is required. It is now actually feedback from the secondary source of the system and the non-ideal property of the components existing in the system which are affecting the attenuation result. The transfer function of the system will be Ho(ie)), shown in eqn (3). In order to eliminate the influence of these two factors, a compensating system with frequency response Hl(i~o ) is used to replace the element e -i'°~ framed with the dotted line as shown in Fig. 3. Then the transfer function of the compensating system will be: -- H M H p H L H 1 H°(i¢~) = 1 + ~ H M H v H L H 1 e-i,o~
(4)
To satisfy the requirement for complete noise attenuation in the duct, the transfer function Ho(i¢o) of the system represented in eqn (4) must be equal to - e -i~'* in the final form. Therefore, we have the transfer function: e - io,~ Hl(ico) = HA(1 -- ~ e-i2,o~)
(5)
where HA = H M H p H L . That is, the constant time delay z adopted in the original system must now be replaced by Hl(ico) represented in eqn (5). Assuming that HA = IHAle-~°'a~ and substituting into eqn (5), we have: e - ito(t - At)
Hl(io9) - IHAI(1 - ~ e-i2o~)
(6)
H~(ioo) can be obtained by a system whose working block diagram is shown in Fig. 4(a), and the transfer function block diagram is shown in Fig. 4(b). Obviously the compensating element H l ( k o ) of the system can eliminate the feedback effect of the secondary source in the upstream direction as well as compensate the non-ideal phase response of the system, but the structure
A monopole system for active attenuation of noise in a duct
193
~ e . _ S e- s( ' ~ - / Y I ; ) I ~ ~
p (a)
(b) Fig. 4.
(a) The working block diagram of the compensating segment H l(io ). (b) The transfer function block diagram of the ACM-System.
L
Power
amplifier White
L
LI
F
"~
noise
generator
(a) Sound level of : 11OdB g > x3
f----_
~.-
- 105 dB
> >
L 0
i°
~ 0
~
200
400 600 800 Frequency (Hz) (b)
95dB
I
1000
Fig. 5. (a) The block diagram for tests of HA(/to); (b) test results of the frequency amplitude response of H^(ito).
Jiluo Zhou, Tielin Shi
194
of the system is somewhat complicated. Experiments have proved that the transfer function of HA can be simplified and obtained by tests according to Fig. 5(a), and test results of [HAl used in this paper are shown in Fig. 5(b). From Fig. 5(b), it can be seen that the amplitude response of [HA[ approximates to a straight line, errors of which are within 2 dB in 1000 Hz bandwidth. Therefore, IHAI can be regarded as a constant A within a limited error. That is H A = A e -i~'a~. Substituting into eqn (6), we have: e - i,,,(, - a~)
Hl(i~) = A(1 - o~e-iz'~)
(7)
The time delay Ar in eqn (7) can be obtained by linear fitting of the phase response curve of H A shown in Fig. 6. The slope of the straight line is the
c
~.o
,.i-
100
I
I
200
300
I ~ l 400
Frequency Fig. 6.
500 (Hz)
600
Test results of the frequency phase response of HA(i~o).
constant time delay Az. The test result in this paper is Ar = 0"31 ms. Then, the transfer function of the compensating sub-system is: e - i~,~r- 0.000 31)
Hl(i~) = A(1 - ~ e-i2'°~)
(8)
The transfer function of Hl(i~o) in eqn (8) can be easily realized by using the integrated circuit. 3.2 The effect of the directional coefficient at of the sound detector on the system characteristics
In the discussions above, the feedback gain coefficient ~' in the transfer function Hl(ico ) shown in Fig. 4(b) has been considered as equal to the directional coefficient ~ of the sound detector, i.e. ~' = ~. In fact, the value of ~' is a constant, and the value of ~ fluctuates with the sound frequency.
A monopole system for active attenuation of noise in a duct
195
Both ~ and ~' are always unequal at different sound frequencies. When ct' q= ~t, eqn (5) becomes: e - i,~ = HA(1 -- 0t' e -i2~'~)
Hi(it°)
(9)
and the transfer function of the whole system is: -
e
-
i,o~
H ° ( i c ° ) = 1 - ~'(1 - cz/cx')e -i~°,
(10)
When 0(= ~, eqn (10) becomes: H o ( iCo ) = _ e - i~,~
This is just the transfer function required by the complete noise attenuation as shown in eqn (2). Otherwise, it can give both the amplitude and the phase errors for the system. The wider the frequency bandwidth is, the greater the fluctuation of~ and the greater the amplitude and phase errors of the system will be. Figure 7 gives the amplitude and the phase errors of the system, when ~' and ~ have different values. The greater the errors of the system, the more Cl{,° - ClL
:
40"1,
OL
CL = 0 . 5
~' ~o 3
•
5
n
I
200
e
I
400 Frequency (a)
~
~-2
800
600 800 Frequency (Hz)
-1
=
600 (Hz)
i
(b)
Fig. 7.
The phase errors and amplitude errors of the ACM-System under different sound frequencies and different values of (ct'-ct)/a, during ct = 0'5.
196
Jiluo Zhou, Tielin Shi
difficult it is to improve the noise attenuation effect of the system in a wide bandwidth. There are three methods of limiting the difference of values between ~' and c~.The first is to select a sound detector with a constant directional coefficient ~. The second is to select an omnidirectional sound detector, whose directional coefficient ~ = 0 , but this is not practicable because of the instability of the system under working conditions. The third is to select a sound detector whose directional coefficient ~ = 1, i.e. the sound detector detects the noise signals only in one direction, and such a detector can be composed of two omnidirectional sound detectors.
3.3 Stability analysis of the ACM-System It is known that the stability problem arises in all closed-loop systems. The necessary and sufficient conditions for self-oscillations of the system are:
[Gl(ico G2(i~o)l > 1 ~ G I ( i ~ ) . G2(i~ ) = 2mz,
forn = 1,2,3 ....
(11)
where Gl(i~ ) is the transfer function of the forward feed channel of the system G2(i~o) is the transfer function of the feedback loop of the system. If one of the conditions represented in e q n ( l l ) is not met, selfoscillations will never arise again. From Fig. 4 we can find that the transfer function of the forward feed channel of the ACM-System is: --IHAle -/°'~
Gl(i°9) = 1 - ~'e -i2c°*
(12)
and the transfer function of the feedback loop of the: ACM-System is:
G2(i~o) = ~ e-i'°~
(13)
Then, the amplitude of the product of above transfer functions Gl(ioJ ) and G2(ico ) is:
IGl(i6o). G2(itn)l =
~IHAI x/1 - 2c( cos 2coz + ~,2
(14)
From eqn (14) we can obtain an absolute stable working condition of the ACM-System as:
~IHAI i x/1 - 2a' cos 2o~z + ~,2
< 1
(15)
A monopole system]br active attenuation of noise in a duct
197
Under the limited condition of cos 2o9r = 1, we get: sinAI
- - < 1 -- ~'
1
(16)
For a practical system for the active attenuation of noise in a duct, the feedback gain coefficient ~' is usually constant and it is c o m m o n to have IHAI~ 1. Substituting these values of ~' and IHAIinto eqn (16), the stability condition of the ACM-System can be obtained as: < 1 - ~'
(17)
With the condition ~ = ~', we get a < 0.5. This condition can be easily satisfied by the ACM-System.
4 TEST R E S U L T S A N D A N A L Y S I S O F T H E A C M - S Y S T E M The experimental set used for testing the working capacity of the system is illustrated in Fig. 8. The sound detector Mic 1 employed is sensitive only to the primary noise and not to the secondary noise or the reflections from the termination of the duct; the sound detector Mic 2 is applied to measure the attenuation results of the system. As to the working process of the whole system, it is easily understood from the block diagram shown in Fig. 8, and it is unnecessary to go into details. Three attenuation tests have been carried out for the ACM-System: the first is the pure tone noise attenuation test; the second is a random noise attenuation test; and the third test is processed with air flowing in the duct. These three tests are described in detail respectively as follows. / 4.1 Pure tone noise attenuation tests o f the A C M - S y s t e m
As we know noise in reality is always of a continuous spectrum. Such a noise spectrum can be regarded as a pile of several pure tone noise spectrums. The Primary
Power
~i
Mic1
MiCci
Second/~ry source
amplifier Sounder box
Signal
generator
Fig. 8,
~ J [ I
~ ~
~
Power amplifier
Spectrum I ana y s e r
I
The sketched experimental set diagram of the ACM-System for tests of its attenuation capacity.
Jiluo Zhou, Tielin Shi
198
c:
.o ¢)
"o
o
1oo
200
Frequency (Hz) Fig. 9.
Test results of the ACM-System for attenuation of a 100 Hz pure tone noise in a duct.
elimination of frequency peaks in the noise spectrum can effectively reduce the sound level of the noise. Figure 9 shows the test result for the attenuation of a 100 Hz pure tone noise in a duct. Curve a is the primary noise spectrum of the system taken before the secondary source is turned on; curve b is the noise spectrum after the secondary source is turned on during the test. (An attenuation test with acoustic material wedges placed just before the termination of a duct has also been processed. The test result shows a similar curve b to that shown in Fig. 9. It means that the much lower reflections from the termination has practically no influence on the test result.) Table 1 shows the test results for the attenuation of pure tone noise with different frequencies. The test results demonstrate that the ACM-System gives good attenuation for the noise in the duct. From Table 1 it can be seen that with increasing noise frequency the attenuation will be decreased. There are mainly two reasons for this: one is due to vibrations of the duct wall, the other is due to the error Ab of the installed distance b' and the theoretical distance b from the detector to the secondary source. The distance error Ab influences the attenuation more at high frequencies than low frequencies. Improvement of the stiffness of the duct wall and accurate control of the distance b from the detector to the TABLE 1 Pure Tone Noise Attenuation Results of the ACM-System During Test in Duct
Sound frequency, Hz Attenuated sound level AL, dB
I00 66
200 45
250 42
300 40
400 38
500 25
800 18
1 000 12
A monopole system for active attenuation of noise in a duct
199
secondary source greatly increases the attenuation of pure tone noise and reduces the noise peaks. 4.2 Random noise attenuation tests of the ACM-System As mentioned above, the main factor which affects the attenuation result is the phase and amplitude error caused by the non-ideal response property of the system components. The linear error can be decreased by the compensating transfer function H~(io~)of the ACM-System. And a specially designed closed type sounder box can be used to improve the radiation capacity of the loudspeaker at low frequencies, and this can result in decreasing the amplitude error. Tests have been performed with 100, 250, 500 and 1000Hz bandwidths respectively. Test results are shown in Fig. 10, and the corresponding
0 I
.__> 2 0 0
I
250
300
I
700
I
750
800
4; > 4;
0
%o -go
55o 6;o 6;o %o %o
> C 0 U~
100
I
I
I
I
I
200
300
400
500
600
Frequency Fig. 10.
0 (Hz)
I
I
I
200
400
600
/
800
I
1000
Test results of the ACM-System for attenuation of the random noise in a duct with different bandwidths.
Jiluo Zhou, Tielin Shi
200
attenuated values of noise are on average 30, 20, 15 and 12 dB respectively. The test results demonstrate that the ACM-System can give a good attenuation on noise of wide as well as narrow bandwidth, although of course, the narrow bandwidth gives better results. Therefore, the ACM-System is superior to any other general monopole system. 4.3 Random noise attenuation tests of the ACM-System with air flowing in a duct When the ACM-System is used with air flowing in a duct, the flowing air will affect the attenuation in the duct; it will influence the sound propagation, and noise detecting accuracy. The former can be neglected when the air flow velocity is low. When the air flow velocity is comparatively high, the effect can be eliminated by adjusting the time delay parameter z in the
"2 O [
> 100 '5
I
I
250
300
I
150
200
200
400
400
II1 "D
>
200
300
I 500
1 600
0
100
Fig. 11.
I
l
I
i
200
300
400
500
i
600 Frequency
I
0 (Hz)
i
200 400
i
I
I
600
800
1000
Test results of the ACM-System for attenuation of the random noise under conditions ofair flow in the duct with different bandwidths.
A monopole system for active attenuation of noise in a duct
201
compensating transfer function Hx(io9). The latter effect can be decreased by using a windshield or a sound wave travelling tube. Test results of the ACM-System obtained under conditions of air flow in the duct are shown in Fig. 11. The air flow velocity in the duct is 20m/s during the test. The sound detector is protected by a windshield in order to improve the testing accuracy. Test results demonstrate that the ACMSystem gives good noise attenuation under conditions of air flow in a duct. The attenuation of the ACM-System, especially for noise of low frequency, approaches the effect of noise in quiet air. From the test results mentioned above, it can be seen that the ACMSystem has the capacity to control the noise in a duct under different working conditions. 5 CONCLUSIONS (1) In modern active attenuation systems for noise in a duct, the monopole system is now under rapid development because of its superior characteristics due to its compact structure, low cost and ease of use. The research focus of the monopole system is still overcoming its weak points and improving the capacity of the system. (2) The amplitude and phase response errors resulting from the non-ideal properties of the system components are the main factors which affect the noise attenuation results of the monopole system. The amplitude response errors of the system, which are mainly produced by the loudspeaker, can be decreased many times by a special acoustic design for improving its radiation capacity. The phase response errors of the system can be corrected by designing a compensating segment Hl(iog) in the system. (3) Through a careful study of the compensating segment Hl(io9 ), it is shown that a rational construction of the sound detector, the power amplifier and the loudspeaker can make the response characteristics of the s y s t e m H A = H M H p H L = A e-i,,a~, where A is constant. This means that the detector, the power amplifier and the loudspeaker are all unnecessary for acquiring a better response. (4) A directional detector introduced in the monopole system can improve the working stability of the system, lfthe directional coefficient a of the sound detector and the feedback gain coefficient a' in the ACM-System are satisfied by eqn (17), i.e. a < 1 - a ' , the system will be fully stable under working conditions. (5) The ACM-System established according to the theory described above can have a better attenuation of noise than that of the other ordinary monopole system. It is also as good as the multipolar active attenuation system.
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Jiluo Zhou, Tielin Shi REFERENCES
1. Eghtesadi, Kh. & Leventhall, H. G., Active attenuation of noise--The monopole system. J. Acoust. Soc. of Am. 71(3) (1982) 608-11. 2. Chaplin, B., The cancellation of repetitive noise and vibration. In Proceedings of Inter-noise 80, Institute of Noise Control Engineering, USA, 1980, pp. 699-702. 3. Chaplin, B., An adaptive wave-form generator for cancelling engine exhaust noise. Report for the Institute of Electronic and Radio Engineers, UK, 1980. 4. Warnaka, G. E. & Tichy, J., Acoustic mixing in active attenuators. In Proceedings oflnter-noise 80, Institute of Noise Control Engineering, USA, 1980, pp. 683-8. 5. Tichy, J. & Warnaka, G. E., Effect of evanescent waves of the active attenuation of sound in ducts. In Proceedings of Inter-noise 83, Institute of Acoustics, UK, 1983, pp. 435-8. 6. Martin, V. & Roure, A., Control of sources for active sound propagation in a duct. In Proceedings of Inter-noise 83, Institute of Acoustics, UK, 1983, pp. 431-4. 7. Ross, C. F., An adaptive digital filter for broadband active sound control, J. Sound Vibr., 80(3) (1982) 381-8.