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The influence of waveguide reflections and system configuration on the performance of an active noise attenuator
Journal of Sound and Vibration (1985) 100(4), 569-579
THE INFLUENCE OF WAVEGUIDE REFLECTIONS AND SYSTEM C O N F I G U R A T I O N O N THE PERFORMANCE OF AN ACTIVE NOISE A T T E N U A T O R R. F. LA FONTAINE AND I. C. SHEPHERD Division of Energy Technology, Commonwealth Scientific and Industrial Research Organization, Melbourne, Australia (Received 16 March 1984, and in revised form 25 August 1984)
Reflections of sound from waveguide terminations can play a major role in the performance of a random noise active attenuator. Their influence depends on the direction reflections propagate along the waveguide and on the configuration of the attenuator. Upstream and downstream reflections are therefore important parameters to be considered when choosing the configuration best suited for a particular application. Theoretical and experimental results are presented, showing that waveguide reflections interact with other major parameters of a random noise attenuator. 1. INTRODUCTION Much theoretical and experimental work has been carried out in the field of active attenuation of noise in waveguides, e.g., air conditioning ducts, chimney stacks and exhaust pipes, and many facets of the subject have been disclosed. Jessel and Mangiante [1], for example, have shown that Huygens principle of wave propagation and the principles o f sound absorption share a common foundation. Swinbanks [2] has described a method o f producing unidirectional sound for the control of noise in ducts; Poole and Leventhall [3] have verified Swinbanks' work by experiment and have shown how errors of sound reproduction limit noise attenuation; Ross [4] has demonstrated an adaptive digital filter for optimizing the performance o f a random noise control system; the influence of imperfect unidirectional acoustic couplers on the effectiveness of a random noise attenuation has been explained in reference [5]. While the above work represents a fraction of that published, there are still numerous questions to be answered before the system designer can confidently predict the performance of an attenuator in diverse practical applications. Thus experimental investigation continued on from the study of reference [5], with the object of revealing limitations specific to various configurations of random noise attenuator. Attenuators of interest were the type comprising one or more microphones for sensing offending plane wave noise in ductwork, and one or more loudspeakers for outputting a cancelling sound; the origin of the "simplest system, patented in 1934 by Lueg [6]. The results of these experiments suggested that reflections of sound from duct terminations could, in some arrangements, significantly reduce the effectiveness of an attenuator, apart from the unstabilizing influence of reflections reported by earlier workers. Subsequently a theoretical model was developed to explain the experimental results and to expose other potential limiting factors. Details and results of the experiments are provided here; however, the principal aim of this paper is to describe theoretically the influence of reflections from duct terminations 569 0022-460x/85/120569+ 11 $03.00/0 O 1985 Academic Press Inc. (London) Limited
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R.F. LA FONTAINE AND I. C. SHEPHERD
and discontinuities on the effectiveness and the power efficiency of a random noise active attenuator. 2. THE EXPERIMENTAL MODEL Experiments were conducted on three noise attenuators which incorporated different arrangements of unidirectional and omnidirectional loudspeakers and microphones. Airflow was not employed in the work reported in this paper because turbulence could not easily be defined or reproduced. Instead, it was accepted that flow would inject pseudosound into microphones to limit the effectiveness o f an attenuator. Nevertheless, such sound is taken into account in the theoretical model. Components of the model were similar to those described in reference [5] and portrayed in Figure 1. The waveguide was a rigid wall duct of cross section 0.244 m square. The Differential amplifier
Microphones(A)h ~(B) ~r
Transversal
Delayft. ~ Amplifiersl I I I (B)T I ( A ) "x SecondaryIoudspeakers/'~
Exponentialhorn termination Noise ~Primory loudspeaker Randomnoise l Amplifier Equalizer generator l" I---1 " D 123
Monitoring microphoneD Spectrum analyzer [S]
Figure 1. The experimental attenuator incorporating unidirectional couplers. Microphone (B) or loudspeaker (B) is disconnected when a coupler is to have an omnidirectional response.
primary noise source comprised eight loudspeakers mounted around the perimeter of the duct, and which were supplied with band-limited random noise to produce plane wave sound at an acoustic pressure o f 100 dB through the range 20-650 Hz. By means o f a third octave band graphic equalizer, the power spectrum of the acoustic pressure was made substantially fiat over the bandwidth, as measured by a microphone located centrally and directly in front of the duct downstream exit. Motional feedback, obtained by utilizing the back emf of the voice coil, was applied to the secondary loudspeakers to improve response, and frequency-dependent amplitude and phase errors of the overall system were minimized by a transversal filter. The technique described in reference [5] was used for setting the transversal filter. The upstream duct termination consisted of a semi-anechoic horn and fan which together had a reflection coefficient varying between 0.6 and 0.05 over the frequency range 30-600 Hz. Downstream termination consisted o f an open-ended duct; from equation (3) o f Davies etal. [7] the modulus of the downstream reflection coefficient was calculated to be between 1 and 0.5 for the frequency range 30-600 Hz. Unidirectional acoustic couplers (microphones and loudspeakers which are sensitive to, or produce, sound propagating only in one direction through a waveguide) employed in the experimental work were of the type contrived by Swinbanks [2]; however the theoretical treatment given in section 3 is applicable to many random noise attenuators where the noise-sensing microphone is located between the loudspeaker and the primary
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ACTIVE NOISE ATTENUATION IN WAVEGLIIDES
noise source. Unidirectional couplers comprised two loudspeakers, two microphones, and delays rL and ~'M- Omnidirectional couplers comprised only the outer-most microphone and loudspeaker of Figure 1. Directional response of the unidirectional loudspeaker was determined by the method described in reference [3]. Response of the unidirectional microphone was estimated by measuring the sensitivity of the microphone to acoustic pressure pulses injected at both ends of the duct. Pulse duration and repetition rate were so selected that the pulse and its reflections were separately distinguishable on a cathode ray oscilloscope. The mean system gain (g) and mean phase error (~), employed later in the discussion, were determined from the adjustments of the system gain trimmer gr and delay rx needed to maximize the attenuation of sound at a number of test frequencies. These adjustments represent residual gain and phase errors which could not be accommodated by the transversal filter. Delay rx primarily enabled matching of the electro-acoustic and duct propagation times to be accomplished. 3. THE THEORETICAL MODEL The theoretical model provides a relationship between the incident and resultant sounds in terms of the attenuator and duct termination parameters. This can be done in the frequency domain in the form of statistical spectra by using the theory of random data. Parameters of the model, shown in Figure 2, are as follows: P, the radiated primary System |ronsfer function
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Figure 2. The theoretical model. P, Primary random noise source; E, extraneous noise source; S, secondary noise source; A, total upstream generated noise; B, total downstream generated noise; C, uncancelled noise; D, waveguide exit noise; g, q~,overall system gain and phase; a . , aM, loudspeaker and microphone back-to-front ratios; Ro, Ru, downstream and upstream reflection coefficients.
random noise; E, extraneous noise due to turbulent pressure fluctuations, unfiltered higher order mode sound and/or electronic noise; A, the total plane wave sound to be controlled, comprising sound P and the sound returned by upstream reflection; S, sound output by the secondary loudspeaker(s) to cancel sound A; C, sound resulting from imperfections of attenuation; B, sound returned through downstream reflection plus sound radiated in the upstream direction by the secondary loudspeaker(s); D, sound escaping from the waveguide; (3, the complex transfer function ofthe entire electro-acoustic system represented here as the system phase error q5 and the total system gain g (when g = 1 the acoustic pressure of the sound radiated by the loudspeaker(s) is the same as that heard by the microphone(s)); Ro and Ru downstream and upstream reflection coefficients; aL and aM, loudspeaker and microphone directivity coefficients expressed as the back-to-front ratio a, or the sensitivity in the backward direction divided by the sensitivity in the forward direction; an ideal unidirectional coupler has a coefficient of a =0, while an omnidirectional coupler has a coefficient of a = 1.
572
R . F . LA FONTAINE A N D 1. C. S H E P H E R D
The analysis refers to all variables in the frequency domain. It is assumed that the propagation times ofthe waveguide and the electro-acoustic equipment have been adjusted to be identical, apart from phase lag or lead errors (r catered for in the following equations. It is further assumed that the primary noise source generates random noise (P), and that the components of sound only correlate where noted in the text. Sound C results from the addition of upstream noise A and secondary sound S. The power spectral density (PSD) of C is given by random data theory according to Newland [8] as so(o,) = sA(o,) + SAd,o) + Ss~(O,) + Ss(o~),
(1)
where S c ( w ) is the PSD of sound C, Sa(oJ) is the PSD of sound A, Ss(a,) is the PSD of sound S, Sas(~O) is the cross spectral density of A and S, and Ssa(W)= S*s(a,). (* denotes the complex conjugate.) Since it is intended that A and S should correlate strongly, Sas(W) and Ssa(a,) are important components in equation (1). Again using the theory of random data, one has SAS(r
= S~A(O) ) = G ( ~ ) S A ( t O
)
Ss --[~M (~o)G(,o)lZSo (o~) + IG(o~)12Sa(oJ) + Io(,o)l=sE
and
(2)
(o,),
(3)
where G(~o) is the complex frequency response of the system relating sound at the microphone to secondary sound S, aM (to) is the complex ratio of downstream to upstream sensitivities of the microphones, Sn(~o) is the PSD of the sound propagating back upstream past the microphones, and SE (w) is the PSD of the combined spurious sounds. Substituting equations (2) and (3) into equation (1) and omitting (w) for clarity gives S C = S A -{- ( G "1- G s : ) S A d l - I ~ ^ , c l 2 s o + [GI2SA + IGI2SE.
(4)
Representing G in equation (4) by - g e i6, where the gain g and r are real and frequency dependent, results in S C = X S a + Ol2fg2SB "~ g 2 S E ,
(5)
where X = 1 - 2 g cos r Sound A is the sum of the primary noise P and reflections of B. Since it can reasonably be assumed that these two components are uncorrelated, SA = Sp + R2vSB.
(6)
Substituting equation (6) into equation (5) gives Sc = XSp + ( X R 2 a 2 , g 2 ) S n + g2S E.
(7)
Similarly, sound B is the sum of the upstream radiation from the secondary source ctLS and reflections from the downstream duct system RDC. Thus, by using equation (3), SB
= a L2 S s + a o2s c
2 2 = a L2( a M g S n + g 2 S A + g 2 S e ) + R D2S c .
(8)
Substituting for SA by using equation (6) and rearranging terms gives So = { a 2 g 2 ( S p + S e ) + R ~ S c } / { 1 - a 2Lg 2( a 2^, + R ~ ) } .
(9)
When equation (9) is substituted into equation (7), Sc is obtained by rearrangement; thus Sc=
X S p + g 2 S e ) { 1 - a ,2g 2(a 2M+ g 2v)} + ct~g2(XR 2 + t~],g 2)(Sp + SE) 1 - a L2g 2(OtM+R~) 2 2 2 2)RD 2 -(XRv+aMg
(10)
ACTIVE
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AT1-ENUATION
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573
When aL, aM, Ru, Ro and SE are all zero, equation (10) reduces to
Sc = XSp = ( 1 - 2 g cos q5+ g2)S~
(11)
which should be familiar to those acquainted with reference [3]. With the downstream termination assumed to be energy conserving, the PSD beyond the termination is
So=(1-R2)So
(12)
When the attenuator is not switched on ( g = O in equation (10)),
Sc = SA=Sp/(1 - R u2R o )2.
(13)
4. RESULTS AND DISCUSSION Because o f the number of parameters in the theoretical model, discussion is limited to the influence of waveguide reflections on three arrangements of acoustic couplers. Parameter values adopted are g = 0 . 9 , ~b = 5 degrees, aM =0"1, and aL=0"18, corresponding approximately to those o f the experimental attenuators. These parameters are normally a function of frequency; however, in the discussion the mean values over the operating frequency range of the attenuators are used. The mean values were determined after the attenuators had been adjusted to minimize the RMS sound pressure at the duct exit with random noise applied to the primary loudspeakers. For clarity, extraneous noise is ignored (SE =0). Insertion loss is used as the measure of attenuator effectiveness, and represents the ratio of sound D at the downstream boundary without attenuation, to the corresponding noise with attenuation: i.e., Doa/Don determined from equations (10), (12) and (13). The quantitative term attenuation has been avoided since by definition [9] attenuation is the ratio of the duct sound power levels measured either side o f a noise control device. Insertion loss and attenuation differ considerably in value for a system which reflects noise back to the source. Of the two measures, insertion loss most closely relates to the reduction o f noise as experienced by the observer. 4.1. O M N I D I R E C T I O N A L LOUDSPEAKER AND UNIDIRECTIONAL MICROPHONE Consider the effect of reflection from the upstream termination in an errorless system ( g = 1, a M = q b = 0 ) in which random noise travelling in the downstream direction is sensed by the microphone and reproduced by the loudspeaker to precisely cancel the noise. The loudspeaker radiates equally in both directions (aL= 1), with some sound reflected back by the upper boundary to augment the primary noise. Assuming the reflected and primary noise are uncorrelated and the downstream reflection coefficient Ro is zero, the mean square sound pressure level incident on the microphone is given by S p ( 1 g2R2)-l. Thus for a system gain of g = 1, the upstream noise level rises by the factor (1 -R2) -~ when the attenuator is switched on. Provided both amplifier and loudspeaker have adequate power ratings, the combined primary and reflected secondary sound will be cancelled completely as before. In an imperfect system, e.g., g = 0.9, the sound level beyond the downstream boundary rises to nearly 23% of the primary sound amplitude as Rv approaches unity. Thus the insertion loss has been reduced from 20 dB, corresponding to a gain of 0.9 to 13 dB in the presence of reflection from the upstream termination. Figure 3 shows the influence of upstream reflection and gain on the waveguide exit noise for a system without phase error.
574
R. F, LA F O N T A I N E A N D I, C. S H E P H E R D I
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0.2
0.4
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0-8
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Figure 3. Ratio of waveguide exit noise level to primary noise level versus upstream reflection coefficient and system gain. a L = I, aM = R o = q5 = O.
Reflection from the downstream termination undoubtedly influences the noise level at the waveguide exit, but has little influence on the attenuator insertion loss (Don/Do.) for small values of Rv, because in this case RD contributes almost equally to the sound levels Don and Do,. However, when Ru approaches unity, the insertion loss is markedly influenced by Ro, and thus Don~Do, ~- ([{(1 + g ) / ( 1 - g ) } - R ~ ] / [ 1 - R ~ ] ) '/2,
(14)
which tends to infinity if the system gain approaches unity or R D approaches unity (and g > 0). In the latter instance an attenuator system would of course not be needed. While the probability of such large values of reflection is unlikely (except at very low frequencies), the extreme case explains the trend predicted in Figure 4. I
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Figure 4. Insertion loss versus downstream reflection ( R o ) for an attenuator with an omnidirectional loudspeak er and a unidirectionl microphone with various values of upstream reflection ( R u ) . a L = 1, a M = 0 . 1 , g = 0-9, q~ = 5 degrees.
Excessive demands on attenuator power output can also accompany a large reflection from the upstream termination, as typified by the secondary source power spectral density Ss plotted in Figure 5. The curves shown are for the ease g = 0.9, so that the relative PSD will be greater with larger gains. For example, the relative PSD is at least 5-5 when
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Figure 5. Relative power spectral density of the secondary source versus downstream reflection (Ro) and upstream reflection (Rv) for a system with an omnidirectional loudspeakerwith a M = 0, g = 0.9, ~b= 5 degrees.
Ru is 0.9 and g = 1. RD has negligible influence on power, provided the attenuator is reasonably effective in reducing the primary noise level: i.e., if sound C is small then only a small amount o f noise is reflected back upstream to augment the primary noise. Figure 6(b) shows the insertion loss achieved by the experimental attenuator, and with the transversal filter set to maximize the reduction of sound at selected frequencies over the bandwidth of operation. A sound level meter reading taken at the exit of the duct indicated 14 dB reduction of noise, about 1 dB less than that predicted in Figure 4 for the equivalent theoretical case (with R o = 0 . 5 to 1 and Ru = 0 . 0 5 to 0.6). 30
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Figure 6. Insertion loss for the experimental attenuators versus frequency. (a) , Unidirectional loudspeaker and microphone; (b) - - - - - , omnidirectional loudspeaker and unidirectional microphone; (c) - - - , unidirectional loudspeaker and omnidirectional microphone.
Performance Was poor at low frequencies compared with that of the two unidirectional coupler system (see Figure 6(a)). Following the theoretical investigation, the cause was traced to distortion in the early stages o f the attenuator electronics--a product of largerthan-normal microphone signal resulting from upstream reflection. Better performance
576
R. IF. LA FONTAINE AND 1. C. SHEPHERD
could be expected with a reduced primary noise level, or preferably with improved circuit design. 4.2. OMNIDIRECTIONAL MICROPHONE AND UNIDIRECTIONAL LOuDsPEAKER A unidirectional loudspeaker which exhibits a small back-to-front ratio over a wide frequency range is difficult to produce; a figure o f - 1 5 dB ( a L = 0 . 1 8 ) could reasonably be expected over a bandwidth of four or more octaves. Thus, as indicated by equation (22) of reference [5], an insertion loss of about 15 dB (for random noise) is the most that could be anticipated with such a coupler. An additional limitation is imposed by downstream reflection as illustrated in Figure 7. Poor performance is predicted for a large downstream reflection coupled with a small upstream reflection. J
1
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18
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16
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14
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c
12
10 0
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0"2
0"4
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0"6
0"8
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Figure 7. Insertion loss versus downstream reflection (RD) for an attenuator with an omnidirectional microphone a n d unidirectional loudspeaker and with various values of upstream reflection ( R u ) . a~f = 1, at =0.18, g =0"9, 4) = 5 degrees.
Suppose g, aM and RD approach unity and R u is zero; in response to primary noise, the loudspeaker outputs the secondary and opposing sound S. Some sound is returned upstream ,(aLS) which is sensed in full by the microphone (a^f = 1) and subsequently reproduced by the loudspeaker. The resulting sound together with any uncancelled noise is recirculated by virtue of the downstream reflection and the omnidirectional microphone, to limit the effectiveness of the attenuator. Additionally, external noise entering the downstream end of the waveguide is sensed by the microphone, although the influence of this noise lessens with increasing values of Ro- Equation (9) is easily modified to incorporate such noise as a parameter o f the theoretical model. The apparent improvement of attenuator performance with increasing reflection from the upstream termination is misleading. For a system employing an omnidirectional microphone and perfect unidirectional loudspeaker, equation (I0) contracts to
s c = ( x s ~ + g2 S~ ) / [ 1 - ( X R ~ + g~) R~,].
(15)
When system errors are reasonably small ( g > 0 . 8 and ~b < I0 degrees) the attenuator insertion loss is given by Don~Do. ~- [(1 - g 2 R 2 ) / X ( 1 - R2R2D)(1 - g)2],/2.
(16)
ACTIVE NOISE ATTENUATION
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577
An increase in insertion loss results if Ru is increased; however, sound leaving the waveguide is not reduced. In effect, an increase in upstream reflectionraises the waveguide noise level while the attenuator is inoperative, but has little influence when the attenuator is switched on. It is therefore essential to remember that Figure 7 represents the ratio Don~Don, which is not directly indicative of the sound level beyond the downstream termination. The performance of an experimental attenuator which incorporated a single microphone and a two-loudspeaker unidirectional secondary source is shown in Figure 6(c). The inferior performance of this arrangement of couplers relative to that shown in Figures 6(a) and (b) in fact stimulated the theoretical work. Parameter values for the experimental attenuator were as1 = 1, CtL= 0"18, g =0"9, and q~= 5 degrees (mean), with upstream and downstream reflection coefficients approaching 0.6 and 1 respectively at low frequencies (below 100 Hz). The theoretical model discloses the nature of the problem as the influence of downstream reflection when an omnidirectional microphone is used; examination of Figure 7 shows that theory predicts poor results for such conditions. An insertion loss of 14 dB maximum is indicated for higher frequencies where the reflection coefficients become smaller, this agreeing with the measured values plotted in Figure 6(c). 4.3. TWO UNIDIRECTIONALCOUPLERS A system in which a unidirectional loudspeaker and a unidirectional microphone are employed is very tolerant of waveguide reflections, at the cost of a larger number of components and a poor secondary source power efficiency for broadband operation (refer to the coupler amplitude response in Figure 2 of reference [5]). Nevertheless, the source efficiency is approximately equal to that of a single loudspeaker system at frequencies close to the centre of the operating band. Figure 8 shows that the performance (Don~Don) can improve in the presence of waveguide reflections with this arrangement of couplers. However, the introduction of a reflecting upstream termination to improve the apparent performance does not reduce the waveguide exit noise, as noted earlier in respect to the omnidirectional microphone system. Additional downstream reflection might be gainfully employed. I
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23
0.8
21
0.6
17
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I 0-4
no
I 0-6
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F i g u r e 8. I n s e r t i o n loss versus d o w n s t r e a m reflection (RD) for a n a t t e n u a t o r w i t h a u n i d i r e c t i o n a l l o u d s p e a k e r a n d m i c r o p h o n e a n d w i t h v a r i o u s v a l u e s o f u p s t r e a m reflection ( R u ) . aL = 0" 18, ct M = 0-1, g = 0-9, ~b = 5 degrees.
578
R.F.
LA FONTAINE
AND
I. C. S H E P H E R D
Considerable overload of the loudspeaker a n d / o r amplifier can arise if termination reflections are large and the system gain is permitted to approach a factor of two or more. Such a condition might occur at the low frequency end of an attenuator's operating range, where the probability o f large reflections is greatest, and where the system gain might not be properly controlled. With a^t aL = ~b = 0 and g = 2 one has S c = Sp/(l -RuRD).2 2 The denominator is seen to approach zero when the product o f R 2 and R 2 approaches unity; thus loudspeaker overload would result. The influence o f overload is not confined to a frequency where the gain is ex(:~ssive, since the amplifier and loudspeaker will not reproduce any frequency correctly in a saturated and non-linear state. Of course, the performance of an attenuator which employs an omnidirectional coupler is even more sensitive to excessive system gains and acoustic reflections. Figure 6(a) shows the insertion loss achieved when two unidirectional couplers ( a L = 0"18 and ot^f = 0.1) were used, and the transversal filter was optimized for this particular arrangement. Degraded performance at the high frequency end of the spectrum was due to ailing loudspeaker response and the influence of a low pass filter which prevented sound reproduction above the design range of the attenuator. Performance at the lower end of the spectrum was mainly limited by a high pass filter which kept loudspeaker cone displacement within recommended limits during the reproduction o f low frequencies. Noise measurements made independently with a sound pressure level meter indicated an insertion loss of 17.5 dB RMS, compared with 17.5-18.5 dB predicted in Figure 8. Such close agreement is not surprising since inaccuracies in estimates of reflection are of little significance (because reflections have minimal influence on the performance o f this type o f attenuator). =
5. CONCLUSIONS In the work presented here, the performances o f experimental random noise attenuators comprising various arrangements of acoustic couplers have been compared. Experiments revealed the complicity of termination reflections with certain combinations of coupler, namely those incorporating an omnidirectional microphone or loudspeaker, in reducing the effectiveness of an attenuator. To understand this behaviour, theoretical work was undertaken and an equation developed which embodies factors representing reflections from waveguide terminations and discontinuities, coupler directional response, uncorrelated extraneous noise, and attenuator gain and phase errors. The major conclusions are as follows. (a) The performance of a random noise attenuator with both a unidirectional loudspeaker and a unidirectional microphone is least affected by reflections. The power efficiency of a unidirectional loudspeaker is small over a wide frequency range; thus the attenuator is best suited to applications where the bandwidth of operation is two octaves or less, or where the attenuator does not have to cope with large sound power levels. However, in situations where upstream reflections are large, this class of attenuator may be the only alternative. (b) An attenuator with an omnidirectional loudspeaker is most sensitive to reflections from the upstream termination. The power dissipation of the attenuator is a function o f reflection, and loudspeaker and amplifier ratings will be greater for large values o f upstream reflection. A knowledge of the coefficient of upstream reflection versus frequency is therefore required before a good estimate of attenuator power output can be made. This will be the case for all active attenuators with an omnidirectionl loudspeaker, whether the secondary sound has been derived from the primary noise, or has been synthesized. (c) Reflections from the exit termination play a significant role in the performance o f an attenuator with an omnidirectional microphone. In view of the small component cost
ACTIVE NOISE AT'FENUATION IN WAVEGUIDES
579
difference b e t w e e n this type o f a t t e n u a t o r a n d that o f (a), it is d o u b t f u l w h e t h e r a n o m n i d i r e c t i o n a l m i c r o p h o n e system w o u l d find practical a p p l i c a t i o n s .
REFERENCES 1. M. J. M. JESSEL and G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390. Active sound absorbers in an air duct. 2. M. A. SWINBANKS 1973 Journal of Sound and Vibration 27, 411-436. The active control of sound propagating in ducts. 3. J. H. B. POOLE and H. G. LEVENTHALL 1976 Journal of Sound and Vibration 49, 257-266. An experimental study of Swinbanks' method of active attenuation of sound in ducts. 4. C. F. R o s s 1982 Journal of Sound and Vibration 80, 381-388. An adaptive digital filter for broadband active sound control. 5. R. F. LA FONTAINE and I. C. SHEPHERD 1983 Journal of Sound and Vibration 91, 351-362. An experimental study of a broadband active attenuator for cancellation of random noise in ducts. 6. P. LUEG 1936 U.S. Patent No. 2043416. Process of silencing sound oscillations. 7. P. O. A. L. DAVIES, J. L. BENTO COELHO and M. BHATrACHARYA 1980 Journal of Sound and Vibration 72, 543-546. Reflection coefficients for an unflanged pipe with flow. 8. O. E. NEWLAND 1975 Random Vibration and Spectral Analysis. London: Longman. 9. L. L. BERANEK 1971 Noise and Vibration Control. New York: McGraw-Hill. See pp. 363-364.
THE INFLUENCE OF WAVEGUIDE REFLECTIONS AND SYSTEM C O N F I G U R A T I O N O N THE PERFORMANCE OF AN ACTIVE NOISE A T T E N U A T O R R. F. LA FONTAINE AND I. C. SHEPHERD Division of Energy Technology, Commonwealth Scientific and Industrial Research Organization, Melbourne, Australia (Received 16 March 1984, and in revised form 25 August 1984)
Reflections of sound from waveguide terminations can play a major role in the performance of a random noise active attenuator. Their influence depends on the direction reflections propagate along the waveguide and on the configuration of the attenuator. Upstream and downstream reflections are therefore important parameters to be considered when choosing the configuration best suited for a particular application. Theoretical and experimental results are presented, showing that waveguide reflections interact with other major parameters of a random noise attenuator. 1. INTRODUCTION Much theoretical and experimental work has been carried out in the field of active attenuation of noise in waveguides, e.g., air conditioning ducts, chimney stacks and exhaust pipes, and many facets of the subject have been disclosed. Jessel and Mangiante [1], for example, have shown that Huygens principle of wave propagation and the principles o f sound absorption share a common foundation. Swinbanks [2] has described a method o f producing unidirectional sound for the control of noise in ducts; Poole and Leventhall [3] have verified Swinbanks' work by experiment and have shown how errors of sound reproduction limit noise attenuation; Ross [4] has demonstrated an adaptive digital filter for optimizing the performance o f a random noise control system; the influence of imperfect unidirectional acoustic couplers on the effectiveness of a random noise attenuation has been explained in reference [5]. While the above work represents a fraction of that published, there are still numerous questions to be answered before the system designer can confidently predict the performance of an attenuator in diverse practical applications. Thus experimental investigation continued on from the study of reference [5], with the object of revealing limitations specific to various configurations of random noise attenuator. Attenuators of interest were the type comprising one or more microphones for sensing offending plane wave noise in ductwork, and one or more loudspeakers for outputting a cancelling sound; the origin of the "simplest system, patented in 1934 by Lueg [6]. The results of these experiments suggested that reflections of sound from duct terminations could, in some arrangements, significantly reduce the effectiveness of an attenuator, apart from the unstabilizing influence of reflections reported by earlier workers. Subsequently a theoretical model was developed to explain the experimental results and to expose other potential limiting factors. Details and results of the experiments are provided here; however, the principal aim of this paper is to describe theoretically the influence of reflections from duct terminations 569 0022-460x/85/120569+ 11 $03.00/0 O 1985 Academic Press Inc. (London) Limited
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R.F. LA FONTAINE AND I. C. SHEPHERD
and discontinuities on the effectiveness and the power efficiency of a random noise active attenuator. 2. THE EXPERIMENTAL MODEL Experiments were conducted on three noise attenuators which incorporated different arrangements of unidirectional and omnidirectional loudspeakers and microphones. Airflow was not employed in the work reported in this paper because turbulence could not easily be defined or reproduced. Instead, it was accepted that flow would inject pseudosound into microphones to limit the effectiveness o f an attenuator. Nevertheless, such sound is taken into account in the theoretical model. Components of the model were similar to those described in reference [5] and portrayed in Figure 1. The waveguide was a rigid wall duct of cross section 0.244 m square. The Differential amplifier
Microphones(A)h ~(B) ~r
Transversal
Delayft. ~ Amplifiersl I I I (B)T I ( A ) "x SecondaryIoudspeakers/'~
Exponentialhorn termination Noise ~Primory loudspeaker Randomnoise l Amplifier Equalizer generator l" I---1 " D 123
Monitoring microphoneD Spectrum analyzer [S]
Figure 1. The experimental attenuator incorporating unidirectional couplers. Microphone (B) or loudspeaker (B) is disconnected when a coupler is to have an omnidirectional response.
primary noise source comprised eight loudspeakers mounted around the perimeter of the duct, and which were supplied with band-limited random noise to produce plane wave sound at an acoustic pressure o f 100 dB through the range 20-650 Hz. By means o f a third octave band graphic equalizer, the power spectrum of the acoustic pressure was made substantially fiat over the bandwidth, as measured by a microphone located centrally and directly in front of the duct downstream exit. Motional feedback, obtained by utilizing the back emf of the voice coil, was applied to the secondary loudspeakers to improve response, and frequency-dependent amplitude and phase errors of the overall system were minimized by a transversal filter. The technique described in reference [5] was used for setting the transversal filter. The upstream duct termination consisted of a semi-anechoic horn and fan which together had a reflection coefficient varying between 0.6 and 0.05 over the frequency range 30-600 Hz. Downstream termination consisted o f an open-ended duct; from equation (3) o f Davies etal. [7] the modulus of the downstream reflection coefficient was calculated to be between 1 and 0.5 for the frequency range 30-600 Hz. Unidirectional acoustic couplers (microphones and loudspeakers which are sensitive to, or produce, sound propagating only in one direction through a waveguide) employed in the experimental work were of the type contrived by Swinbanks [2]; however the theoretical treatment given in section 3 is applicable to many random noise attenuators where the noise-sensing microphone is located between the loudspeaker and the primary
571
ACTIVE NOISE ATTENUATION IN WAVEGLIIDES
noise source. Unidirectional couplers comprised two loudspeakers, two microphones, and delays rL and ~'M- Omnidirectional couplers comprised only the outer-most microphone and loudspeaker of Figure 1. Directional response of the unidirectional loudspeaker was determined by the method described in reference [3]. Response of the unidirectional microphone was estimated by measuring the sensitivity of the microphone to acoustic pressure pulses injected at both ends of the duct. Pulse duration and repetition rate were so selected that the pulse and its reflections were separately distinguishable on a cathode ray oscilloscope. The mean system gain (g) and mean phase error (~), employed later in the discussion, were determined from the adjustments of the system gain trimmer gr and delay rx needed to maximize the attenuation of sound at a number of test frequencies. These adjustments represent residual gain and phase errors which could not be accommodated by the transversal filter. Delay rx primarily enabled matching of the electro-acoustic and duct propagation times to be accomplished. 3. THE THEORETICAL MODEL The theoretical model provides a relationship between the incident and resultant sounds in terms of the attenuator and duct termination parameters. This can be done in the frequency domain in the form of statistical spectra by using the theory of random data. Parameters of the model, shown in Figure 2, are as follows: P, the radiated primary System |ronsfer function
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.
.
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Figure 2. The theoretical model. P, Primary random noise source; E, extraneous noise source; S, secondary noise source; A, total upstream generated noise; B, total downstream generated noise; C, uncancelled noise; D, waveguide exit noise; g, q~,overall system gain and phase; a . , aM, loudspeaker and microphone back-to-front ratios; Ro, Ru, downstream and upstream reflection coefficients.
random noise; E, extraneous noise due to turbulent pressure fluctuations, unfiltered higher order mode sound and/or electronic noise; A, the total plane wave sound to be controlled, comprising sound P and the sound returned by upstream reflection; S, sound output by the secondary loudspeaker(s) to cancel sound A; C, sound resulting from imperfections of attenuation; B, sound returned through downstream reflection plus sound radiated in the upstream direction by the secondary loudspeaker(s); D, sound escaping from the waveguide; (3, the complex transfer function ofthe entire electro-acoustic system represented here as the system phase error q5 and the total system gain g (when g = 1 the acoustic pressure of the sound radiated by the loudspeaker(s) is the same as that heard by the microphone(s)); Ro and Ru downstream and upstream reflection coefficients; aL and aM, loudspeaker and microphone directivity coefficients expressed as the back-to-front ratio a, or the sensitivity in the backward direction divided by the sensitivity in the forward direction; an ideal unidirectional coupler has a coefficient of a =0, while an omnidirectional coupler has a coefficient of a = 1.
572
R . F . LA FONTAINE A N D 1. C. S H E P H E R D
The analysis refers to all variables in the frequency domain. It is assumed that the propagation times ofthe waveguide and the electro-acoustic equipment have been adjusted to be identical, apart from phase lag or lead errors (r catered for in the following equations. It is further assumed that the primary noise source generates random noise (P), and that the components of sound only correlate where noted in the text. Sound C results from the addition of upstream noise A and secondary sound S. The power spectral density (PSD) of C is given by random data theory according to Newland [8] as so(o,) = sA(o,) + SAd,o) + Ss~(O,) + Ss(o~),
(1)
where S c ( w ) is the PSD of sound C, Sa(oJ) is the PSD of sound A, Ss(a,) is the PSD of sound S, Sas(~O) is the cross spectral density of A and S, and Ssa(W)= S*s(a,). (* denotes the complex conjugate.) Since it is intended that A and S should correlate strongly, Sas(W) and Ssa(a,) are important components in equation (1). Again using the theory of random data, one has SAS(r
= S~A(O) ) = G ( ~ ) S A ( t O
)
Ss --[~M (~o)G(,o)lZSo (o~) + IG(o~)12Sa(oJ) + Io(,o)l=sE
and
(2)
(o,),
(3)
where G(~o) is the complex frequency response of the system relating sound at the microphone to secondary sound S, aM (to) is the complex ratio of downstream to upstream sensitivities of the microphones, Sn(~o) is the PSD of the sound propagating back upstream past the microphones, and SE (w) is the PSD of the combined spurious sounds. Substituting equations (2) and (3) into equation (1) and omitting (w) for clarity gives S C = S A -{- ( G "1- G s : ) S A d l - I ~ ^ , c l 2 s o + [GI2SA + IGI2SE.
(4)
Representing G in equation (4) by - g e i6, where the gain g and r are real and frequency dependent, results in S C = X S a + Ol2fg2SB "~ g 2 S E ,
(5)
where X = 1 - 2 g cos r Sound A is the sum of the primary noise P and reflections of B. Since it can reasonably be assumed that these two components are uncorrelated, SA = Sp + R2vSB.
(6)
Substituting equation (6) into equation (5) gives Sc = XSp + ( X R 2 a 2 , g 2 ) S n + g2S E.
(7)
Similarly, sound B is the sum of the upstream radiation from the secondary source ctLS and reflections from the downstream duct system RDC. Thus, by using equation (3), SB
= a L2 S s + a o2s c
2 2 = a L2( a M g S n + g 2 S A + g 2 S e ) + R D2S c .
(8)
Substituting for SA by using equation (6) and rearranging terms gives So = { a 2 g 2 ( S p + S e ) + R ~ S c } / { 1 - a 2Lg 2( a 2^, + R ~ ) } .
(9)
When equation (9) is substituted into equation (7), Sc is obtained by rearrangement; thus Sc=
X S p + g 2 S e ) { 1 - a ,2g 2(a 2M+ g 2v)} + ct~g2(XR 2 + t~],g 2)(Sp + SE) 1 - a L2g 2(OtM+R~) 2 2 2 2)RD 2 -(XRv+aMg
(10)
ACTIVE
NOISE
AT1-ENUATION
IN WAVEGUIDES
573
When aL, aM, Ru, Ro and SE are all zero, equation (10) reduces to
Sc = XSp = ( 1 - 2 g cos q5+ g2)S~
(11)
which should be familiar to those acquainted with reference [3]. With the downstream termination assumed to be energy conserving, the PSD beyond the termination is
So=(1-R2)So
(12)
When the attenuator is not switched on ( g = O in equation (10)),
Sc = SA=Sp/(1 - R u2R o )2.
(13)
4. RESULTS AND DISCUSSION Because o f the number of parameters in the theoretical model, discussion is limited to the influence of waveguide reflections on three arrangements of acoustic couplers. Parameter values adopted are g = 0 . 9 , ~b = 5 degrees, aM =0"1, and aL=0"18, corresponding approximately to those o f the experimental attenuators. These parameters are normally a function of frequency; however, in the discussion the mean values over the operating frequency range of the attenuators are used. The mean values were determined after the attenuators had been adjusted to minimize the RMS sound pressure at the duct exit with random noise applied to the primary loudspeakers. For clarity, extraneous noise is ignored (SE =0). Insertion loss is used as the measure of attenuator effectiveness, and represents the ratio of sound D at the downstream boundary without attenuation, to the corresponding noise with attenuation: i.e., Doa/Don determined from equations (10), (12) and (13). The quantitative term attenuation has been avoided since by definition [9] attenuation is the ratio of the duct sound power levels measured either side o f a noise control device. Insertion loss and attenuation differ considerably in value for a system which reflects noise back to the source. Of the two measures, insertion loss most closely relates to the reduction o f noise as experienced by the observer. 4.1. O M N I D I R E C T I O N A L LOUDSPEAKER AND UNIDIRECTIONAL MICROPHONE Consider the effect of reflection from the upstream termination in an errorless system ( g = 1, a M = q b = 0 ) in which random noise travelling in the downstream direction is sensed by the microphone and reproduced by the loudspeaker to precisely cancel the noise. The loudspeaker radiates equally in both directions (aL= 1), with some sound reflected back by the upper boundary to augment the primary noise. Assuming the reflected and primary noise are uncorrelated and the downstream reflection coefficient Ro is zero, the mean square sound pressure level incident on the microphone is given by S p ( 1 g2R2)-l. Thus for a system gain of g = 1, the upstream noise level rises by the factor (1 -R2) -~ when the attenuator is switched on. Provided both amplifier and loudspeaker have adequate power ratings, the combined primary and reflected secondary sound will be cancelled completely as before. In an imperfect system, e.g., g = 0.9, the sound level beyond the downstream boundary rises to nearly 23% of the primary sound amplitude as Rv approaches unity. Thus the insertion loss has been reduced from 20 dB, corresponding to a gain of 0.9 to 13 dB in the presence of reflection from the upstream termination. Figure 3 shows the influence of upstream reflection and gain on the waveguide exit noise for a system without phase error.
574
R. F, LA F O N T A I N E A N D I, C. S H E P H E R D I
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0 0
0.2
0.4
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0-8
1-0
Ru
Figure 3. Ratio of waveguide exit noise level to primary noise level versus upstream reflection coefficient and system gain. a L = I, aM = R o = q5 = O.
Reflection from the downstream termination undoubtedly influences the noise level at the waveguide exit, but has little influence on the attenuator insertion loss (Don/Do.) for small values of Rv, because in this case RD contributes almost equally to the sound levels Don and Do,. However, when Ru approaches unity, the insertion loss is markedly influenced by Ro, and thus Don~Do, ~- ([{(1 + g ) / ( 1 - g ) } - R ~ ] / [ 1 - R ~ ] ) '/2,
(14)
which tends to infinity if the system gain approaches unity or R D approaches unity (and g > 0). In the latter instance an attenuator system would of course not be needed. While the probability of such large values of reflection is unlikely (except at very low frequencies), the extreme case explains the trend predicted in Figure 4. I
I
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0-4 0"6
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08
I-0
Figure 4. Insertion loss versus downstream reflection ( R o ) for an attenuator with an omnidirectional loudspeak er and a unidirectionl microphone with various values of upstream reflection ( R u ) . a L = 1, a M = 0 . 1 , g = 0-9, q~ = 5 degrees.
Excessive demands on attenuator power output can also accompany a large reflection from the upstream termination, as typified by the secondary source power spectral density Ss plotted in Figure 5. The curves shown are for the ease g = 0.9, so that the relative PSD will be greater with larger gains. For example, the relative PSD is at least 5-5 when
ACTIVE
NOISE i
i
ATTENUATION i
i
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i
i
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Figure 5. Relative power spectral density of the secondary source versus downstream reflection (Ro) and upstream reflection (Rv) for a system with an omnidirectional loudspeakerwith a M = 0, g = 0.9, ~b= 5 degrees.
Ru is 0.9 and g = 1. RD has negligible influence on power, provided the attenuator is reasonably effective in reducing the primary noise level: i.e., if sound C is small then only a small amount o f noise is reflected back upstream to augment the primary noise. Figure 6(b) shows the insertion loss achieved by the experimental attenuator, and with the transversal filter set to maximize the reduction of sound at selected frequencies over the bandwidth of operation. A sound level meter reading taken at the exit of the duct indicated 14 dB reduction of noise, about 1 dB less than that predicted in Figure 4 for the equivalent theoretical case (with R o = 0 . 5 to 1 and Ru = 0 . 0 5 to 0.6). 30
i
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,
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200
300
400
500
600
Frequency
700
(Hz)
Figure 6. Insertion loss for the experimental attenuators versus frequency. (a) , Unidirectional loudspeaker and microphone; (b) - - - - - , omnidirectional loudspeaker and unidirectional microphone; (c) - - - , unidirectional loudspeaker and omnidirectional microphone.
Performance Was poor at low frequencies compared with that of the two unidirectional coupler system (see Figure 6(a)). Following the theoretical investigation, the cause was traced to distortion in the early stages o f the attenuator electronics--a product of largerthan-normal microphone signal resulting from upstream reflection. Better performance
576
R. IF. LA FONTAINE AND 1. C. SHEPHERD
could be expected with a reduced primary noise level, or preferably with improved circuit design. 4.2. OMNIDIRECTIONAL MICROPHONE AND UNIDIRECTIONAL LOuDsPEAKER A unidirectional loudspeaker which exhibits a small back-to-front ratio over a wide frequency range is difficult to produce; a figure o f - 1 5 dB ( a L = 0 . 1 8 ) could reasonably be expected over a bandwidth of four or more octaves. Thus, as indicated by equation (22) of reference [5], an insertion loss of about 15 dB (for random noise) is the most that could be anticipated with such a coupler. An additional limitation is imposed by downstream reflection as illustrated in Figure 7. Poor performance is predicted for a large downstream reflection coupled with a small upstream reflection. J
1
|
I
18
J
16
o
14
o~
c
12
10 0
i ! i~
0"2
0"4
%
0"6
0"8
1"0
Figure 7. Insertion loss versus downstream reflection (RD) for an attenuator with an omnidirectional microphone a n d unidirectional loudspeaker and with various values of upstream reflection ( R u ) . a~f = 1, at =0.18, g =0"9, 4) = 5 degrees.
Suppose g, aM and RD approach unity and R u is zero; in response to primary noise, the loudspeaker outputs the secondary and opposing sound S. Some sound is returned upstream ,(aLS) which is sensed in full by the microphone (a^f = 1) and subsequently reproduced by the loudspeaker. The resulting sound together with any uncancelled noise is recirculated by virtue of the downstream reflection and the omnidirectional microphone, to limit the effectiveness of the attenuator. Additionally, external noise entering the downstream end of the waveguide is sensed by the microphone, although the influence of this noise lessens with increasing values of Ro- Equation (9) is easily modified to incorporate such noise as a parameter o f the theoretical model. The apparent improvement of attenuator performance with increasing reflection from the upstream termination is misleading. For a system employing an omnidirectional microphone and perfect unidirectional loudspeaker, equation (I0) contracts to
s c = ( x s ~ + g2 S~ ) / [ 1 - ( X R ~ + g~) R~,].
(15)
When system errors are reasonably small ( g > 0 . 8 and ~b < I0 degrees) the attenuator insertion loss is given by Don~Do. ~- [(1 - g 2 R 2 ) / X ( 1 - R2R2D)(1 - g)2],/2.
(16)
ACTIVE NOISE ATTENUATION
IN WAVEGUIDES
577
An increase in insertion loss results if Ru is increased; however, sound leaving the waveguide is not reduced. In effect, an increase in upstream reflectionraises the waveguide noise level while the attenuator is inoperative, but has little influence when the attenuator is switched on. It is therefore essential to remember that Figure 7 represents the ratio Don~Don, which is not directly indicative of the sound level beyond the downstream termination. The performance of an experimental attenuator which incorporated a single microphone and a two-loudspeaker unidirectional secondary source is shown in Figure 6(c). The inferior performance of this arrangement of couplers relative to that shown in Figures 6(a) and (b) in fact stimulated the theoretical work. Parameter values for the experimental attenuator were as1 = 1, CtL= 0"18, g =0"9, and q~= 5 degrees (mean), with upstream and downstream reflection coefficients approaching 0.6 and 1 respectively at low frequencies (below 100 Hz). The theoretical model discloses the nature of the problem as the influence of downstream reflection when an omnidirectional microphone is used; examination of Figure 7 shows that theory predicts poor results for such conditions. An insertion loss of 14 dB maximum is indicated for higher frequencies where the reflection coefficients become smaller, this agreeing with the measured values plotted in Figure 6(c). 4.3. TWO UNIDIRECTIONALCOUPLERS A system in which a unidirectional loudspeaker and a unidirectional microphone are employed is very tolerant of waveguide reflections, at the cost of a larger number of components and a poor secondary source power efficiency for broadband operation (refer to the coupler amplitude response in Figure 2 of reference [5]). Nevertheless, the source efficiency is approximately equal to that of a single loudspeaker system at frequencies close to the centre of the operating band. Figure 8 shows that the performance (Don~Don) can improve in the presence of waveguide reflections with this arrangement of couplers. However, the introduction of a reflecting upstream termination to improve the apparent performance does not reduce the waveguide exit noise, as noted earlier in respect to the omnidirectional microphone system. Additional downstream reflection might be gainfully employed. I
I
i
23
0.8
21
0.6
17
0
I 0"2
I 0-4
no
I 0-6
I 0-8
1-0
F i g u r e 8. I n s e r t i o n loss versus d o w n s t r e a m reflection (RD) for a n a t t e n u a t o r w i t h a u n i d i r e c t i o n a l l o u d s p e a k e r a n d m i c r o p h o n e a n d w i t h v a r i o u s v a l u e s o f u p s t r e a m reflection ( R u ) . aL = 0" 18, ct M = 0-1, g = 0-9, ~b = 5 degrees.
578
R.F.
LA FONTAINE
AND
I. C. S H E P H E R D
Considerable overload of the loudspeaker a n d / o r amplifier can arise if termination reflections are large and the system gain is permitted to approach a factor of two or more. Such a condition might occur at the low frequency end of an attenuator's operating range, where the probability o f large reflections is greatest, and where the system gain might not be properly controlled. With a^t aL = ~b = 0 and g = 2 one has S c = Sp/(l -RuRD).2 2 The denominator is seen to approach zero when the product o f R 2 and R 2 approaches unity; thus loudspeaker overload would result. The influence o f overload is not confined to a frequency where the gain is ex(:~ssive, since the amplifier and loudspeaker will not reproduce any frequency correctly in a saturated and non-linear state. Of course, the performance of an attenuator which employs an omnidirectional coupler is even more sensitive to excessive system gains and acoustic reflections. Figure 6(a) shows the insertion loss achieved when two unidirectional couplers ( a L = 0"18 and ot^f = 0.1) were used, and the transversal filter was optimized for this particular arrangement. Degraded performance at the high frequency end of the spectrum was due to ailing loudspeaker response and the influence of a low pass filter which prevented sound reproduction above the design range of the attenuator. Performance at the lower end of the spectrum was mainly limited by a high pass filter which kept loudspeaker cone displacement within recommended limits during the reproduction o f low frequencies. Noise measurements made independently with a sound pressure level meter indicated an insertion loss of 17.5 dB RMS, compared with 17.5-18.5 dB predicted in Figure 8. Such close agreement is not surprising since inaccuracies in estimates of reflection are of little significance (because reflections have minimal influence on the performance o f this type o f attenuator). =
5. CONCLUSIONS In the work presented here, the performances o f experimental random noise attenuators comprising various arrangements of acoustic couplers have been compared. Experiments revealed the complicity of termination reflections with certain combinations of coupler, namely those incorporating an omnidirectional microphone or loudspeaker, in reducing the effectiveness of an attenuator. To understand this behaviour, theoretical work was undertaken and an equation developed which embodies factors representing reflections from waveguide terminations and discontinuities, coupler directional response, uncorrelated extraneous noise, and attenuator gain and phase errors. The major conclusions are as follows. (a) The performance of a random noise attenuator with both a unidirectional loudspeaker and a unidirectional microphone is least affected by reflections. The power efficiency of a unidirectional loudspeaker is small over a wide frequency range; thus the attenuator is best suited to applications where the bandwidth of operation is two octaves or less, or where the attenuator does not have to cope with large sound power levels. However, in situations where upstream reflections are large, this class of attenuator may be the only alternative. (b) An attenuator with an omnidirectional loudspeaker is most sensitive to reflections from the upstream termination. The power dissipation of the attenuator is a function o f reflection, and loudspeaker and amplifier ratings will be greater for large values o f upstream reflection. A knowledge of the coefficient of upstream reflection versus frequency is therefore required before a good estimate of attenuator power output can be made. This will be the case for all active attenuators with an omnidirectionl loudspeaker, whether the secondary sound has been derived from the primary noise, or has been synthesized. (c) Reflections from the exit termination play a significant role in the performance o f an attenuator with an omnidirectional microphone. In view of the small component cost
ACTIVE NOISE AT'FENUATION IN WAVEGUIDES
579
difference b e t w e e n this type o f a t t e n u a t o r a n d that o f (a), it is d o u b t f u l w h e t h e r a n o m n i d i r e c t i o n a l m i c r o p h o n e system w o u l d find practical a p p l i c a t i o n s .
REFERENCES 1. M. J. M. JESSEL and G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390. Active sound absorbers in an air duct. 2. M. A. SWINBANKS 1973 Journal of Sound and Vibration 27, 411-436. The active control of sound propagating in ducts. 3. J. H. B. POOLE and H. G. LEVENTHALL 1976 Journal of Sound and Vibration 49, 257-266. An experimental study of Swinbanks' method of active attenuation of sound in ducts. 4. C. F. R o s s 1982 Journal of Sound and Vibration 80, 381-388. An adaptive digital filter for broadband active sound control. 5. R. F. LA FONTAINE and I. C. SHEPHERD 1983 Journal of Sound and Vibration 91, 351-362. An experimental study of a broadband active attenuator for cancellation of random noise in ducts. 6. P. LUEG 1936 U.S. Patent No. 2043416. Process of silencing sound oscillations. 7. P. O. A. L. DAVIES, J. L. BENTO COELHO and M. BHATrACHARYA 1980 Journal of Sound and Vibration 72, 543-546. Reflection coefficients for an unflanged pipe with flow. 8. O. E. NEWLAND 1975 Random Vibration and Spectral Analysis. London: Longman. 9. L. L. BERANEK 1971 Noise and Vibration Control. New York: McGraw-Hill. See pp. 363-364.