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An experimental study of swinbanks' method of active attenuation of sound in ducts
Journal of Sound and Vibration (1976) 49(2), 257-266
AN EXPERIMENTAL OF ACTIVE
STUDY
ATTENUATION
OF SWINBANKS’ OF SOUND
METHOD
IN DUCTS
J. H. B. POOLE AND H. G. LEVENTHALL Department of Physics, Chelsea College, Pulton Place, London S W6 SPR, England (Received 3 May 1976, and in revisedform 29 July 1976)
The concept of attenuating sound propagating down a duct by means of an antiphase copy of the sound was first put forward forty years ago, but to date no practical system has been produced. The principal problem is to introduce the antiphase signal in such a way that it propagates only in the direction of propagation of the original sound and to ensure that this property can be maintained over a useful frequency range. A suitable arrangement of secondary sound sources has recently been proposed by Swinbanks and this paper describes experiments carried out in an attempt to realise a practical system. A unidirectional array of secondary sources has been successfully constructed around a rectangular duct, using loudspeaker drive units and electronic delays. Sound propagating in the direction of these sources was sampled and a control signal applied to the sources which in turn acted to significantly reduce the amplitude of the sound. Pure tones at frequencies around 150 Hz have been attenuated by more than 50 dB but results with band-limited noise have been less successful. Further work is suggested which should result in a device having significant advantages over conventional splitter silencers, at low frequencies. 1. INTRODUCTION Existing methods of attenuating sound propagating in ducts include absorptive silencers, lined ducts, and plenum chambers. The first two techniques work well for sounds of medium and high frequencies, but they are very inefficient at low frequencies. Plenum chambers designed to work at low frequencies tend to be bulky. The technique of active absorption, in which cancellation is used to reduce sound fields, provides a means of achieving high degrees of attenuation at low frequencies as well as at the higher frequencies. The size and pressure drops are also less than with passive systems. Previous recent work in the field [l-3], has concentrated on attempts to achieve high absorption at discrete frequencies. The present work draws largely on theory developed by Swinbanks [4], and is aimed at achieving a significant degree of absorption over a band of frequencies. The signal which interferes with the unwanted noise is derived directly from the sound field in the duct so that it becomes possible to operate on complex “real life” waveforms.
2. REQUIREMENTS
FOR ACTIVE
ATTENUATION
Figure 1 is a sketch of an idealised active sound absorber for an air duct. Noise propagating down the duct is detected by a microphone which supplies the information necessary for the operation of a system of secondary noise sources. These secondary sources generate a sound pressure field in antiphase with the noise from the original source, effectively silencing the air stream. In an ideal duct, all frequencies below the first cut-off frequency of the duct must propagate as plane waves. Thus it would appear that a single loudspeaker mounted on the duct wall 25-l
258
J. H. B. POOLE AND H. G. LEVENTHALL
~w Figure 1. A simple representation of an active absorber.
would be sufficient to create an accurate copy of the noise signal, the required phase being obtained by suitably delaying the signal from the microphone. Such a system was the subject of a patent in 1936 [5], and a recent experiment carried out by the authors achieved a reduction in the downstream’sound level of a single frequency tone of about 50 dB by this means. (Throughout this paper reference is made to cancellation downstream of the noise source. Similar techniques can be applied to cancellation upstream of a source.) The loudspeaker does not exchange energy with the noise field in the duct, but is analogous to an impedance change and, instead of being absorbed, the noise is effectively reflected back up the duct. A simple theory based on Young’s theory of interference and Huygen’s Principle has been developed by Jesse1 [I]. He has shown that if an arrangement of secondary sources can be found, which in the absence of any incident sound wave can generate a sound wave that propagates only in the downstream direction, then the incident wave will not be reflected, but absorbed by the sources. One suitable arrangement consists of a combination of a monopole and a dipole source. Experiments carried out by Jesse1 and others [l-3], using this combination, have achieved attenuations of up to 70 dB at discrete frequencies. In the Swinbanks arrangement spaced arrays of monopole sources are used to achieve unidirectionality. Both systems require electronic delays and electronic filtering to achieve satisfactory operation over a useful range of frequencies. A weakness shared by both systems is that they only operate on sounds in the plane-wave mode. Thus a practical duct silencing system will need to retain passive absorption methods for the higher order modes (which are relatively easy to attenuate in this way) while using active methods to remove the lowest frequencies which are more difficult to remove by conventional means. 3. SUMMARY
OF SWINBANKS’
WORK
One secondary source arrangement derived by Swinbanks consists of a pair of source “rings” mounted on the duct walls as shown schematically in Figure 2. Each ring consists of four loudspeaker units, one mounted centrally on each face of the (rectangular) duct. If the source strength of the upstream ring at time t is ml(t) and that of the downstream ring is mz(t) then Swinbanks shows that there will be zero output from the combination of the two rings in the upstream direction if q(t)
= -mz(t - b/c&
- M)),
(1)
where b is the ring spacing, c0 is the velocity of sound in still air, and M is the Mach number of the air-flow in the duct. (A positive value of M denotes flow from left to right in Figure 1.) Thus ml must have the same amplitude as mZ but be inverted, and delayed by b/cO(l - M) relative to m2. There will, however, be an output (D) in the downstream direction which will depend on the angular frequency (0) according to the expression D a
sin (0.42) cos w{t - (x - b)/c,( 1 + M) - 2,/2},
(2)
ACTIVEATTENUATION IN DUCTS
259
CL$f?Q ,&--_-_____y
_____,7L-_-___--
Figure 2. The dispositionof secondarysources around a rectangular duct to form a 2-ring array.
n/
277/r, 2%
To
%I
41r/r, PW”,’ 5 4wm m
Angular frequency, w
Figure 3. The downstreamamplitude-frequencyresponseof the 2-ringarray. where z0 = 2b/c,(l - W). This expression is plotted in Figure 3. The shape of this curve is by no means ideal for the purpose of broad-band active absorption, but experiments in which this arrangement has been used have produced some useful results. By adding a further ring source downstream of the other two the useful frequency range can be extended, but since no experiments were carried out with this arrangement it will not be considered here. Details will be found in reference [4]. The reason for the use of ring sources is not given in the theory above, but is considered in detail in the original paper. If all frequencies of interest lie below o,, the lowest cut-off frequency of the duct, then a single loudspeaker unit at each position, x = 0 and x = b, will suffice. Above this frequency, however, a single unit will excite transverse modes in the duct, as well as the plane wave mode, even if the primary disturbance is entirely in the plane wave mode. By using a ring source, the lowest frequency at which transverse modes can be excited becomes 2.8 o, for a square duct. For a circular duct, three equally spaced sources are used, and the factor is 2.1 0,. 4. OTHER SYSTEM REQUIREMENTS Apart from the secondary source array itself, the other necessary components of the system shown in Figure 1 are the microphone and the electronics necessary to convert the microphone signal into a suitable form for driving the secondary sources. The requirements for the processing section are to delay the microphone signal to allow the detected wavefront time to progress to the cancellation sources, to realise equation (I), to shape the response of the input to the To secondary sources
Delay Microphone
TI
Figure 4. Block diagram of a practical active absorber in which a 2-ring secondary source array is used.
J. H. B. POOLE AND H. G. LEVENTHALL
260
secondary sources to compensate in part for the amplitude-frequency response of the source array, and to amplify the processed signal to the required level. These functions are shown in block diagram form in Figure 4. The requirements for the microphone are that there should be no phase shift between the pressure input and the electrical output, or alternatively that any phase shift should be directly proportional to frequency and thus be equivalent to a simple delay. With a perfectly unidirectional secondary source array there should, in theory, be no problem of feedback between the sources and the microphone. In practice, however, there is a danger of feedback from a source array that is not fully directional and also from reflections of imperfectly cancelled noise from further downstream. It is thus desirable to use a microphone that is only sensitive to sound travelling downstream. Such a microphone could be constructed in a fashion similar to that for the unidirectional source. This would consist of two (or three) rings of microphones, the output from the downstream ring(s) being inverted and delayed before being added to the output from the upstream ring. This array would have a similar amplitude/frequency response to that of the secondary source array (Figure 3). For further details of a unidirectional microphone the reader is referred to section 5.2 of reference [4]. 5. THE CONCEPT
OF AN ATTENUATION
COEFFICIENT
For total cancellation of the incident sound downstream of the attenuator the secondary source array must produce a perfect antiphase copy of this sound field. In practice there are likely to be small errors in the phase and amplitude of the cancellation signal and the effectiveness of the system under these conditions can be expressed in terms of an “attenuation coefficient”, A, defined by A = lOlog,,Pf/P& where PI is the acoustic pressure of the sound in zone 1 and P3 the pressure in zone 3 of Figure 1. Alternatively A = 2010g~o(l~pl/l~p
+PI)
where pp is the acoustic pressure at a point in zone 3 due to the primary the pressure at the same point due to the secondary sources.
noise source and p”
s Figure 5. Vector diagram for the calculation of the attenuation coefficient. If, for example, pp = A cos~t, then complete cancellation will be achieved if p” = -A cos wt. If one supposes that, instead, p” turns out to be -Bcos(ot + 6rp), then the value of A can be calculated by vector addition, as shown in Figure 5, to be Ipp + @I = z/W
- B cos 6~)~ +
(B sin &)‘I
=
Az/l 1 -
2(R/A) cos &p + B2/A21
TABLE 1
Theoretical
attenuation coefficient (dB) achievable various conditions of imperfection
under
B/A
i&p = 0”
sq5= 1”
l&$=3”
sfj = 10’
1 1.005 1.05 1.1
4: 26 20
35 33.7 25.7 19.8
25.5 25-4 22.6 18.6
15.2 15.1 14.2 13.6
ACTIVE ATTENUATION
(time variation
being ignored).
Since lppl = A, the attenuation
A = -lOlog,O(jl
261
IN DUCTS
- 2(B/A)cos@
coefficient
becomes
+ B2/A2\}.
(3)
It should be noted that any phase difference will worsen the attenuation coefficient. Thus i$ is not possible to compensate for changes in B/A with changes in hp. To illustrate the effect of small errors on the attenuation obtained, a few values of A calculated by using equation (3) are given in Table 1.
6. THE EXPERIMENTAL
SYSTEM
The experimental arrangement is sketched in Figure 6. The primary source is represented by a loudspeaker at one end of the duct, the other end of which is left open. The outer sections are each 2.5 metres long and made of 25 mm blockboard, while the central section is 0.75 metres long, of similar construction but with a “perspex” front. The microphones may be moved at will in the duct by means of an arrangement of strings and pulleys. Au--------------I ’
Secondary
sources
(Not to scale)
Figure 6. A sketch of the experimental apparatus.
The secondary source rings each consist of four Grampian SP 25 horn driver units, one mounted centrally on each face of the duct as shown, and connected in series. The ring spacing was chosen to give a “centre frequency” (w,, on Figure 3) of about 150 Hz, the actual dimensions arrived at giving 162 Hz. It was hoped that with this centre frequency useful results would be obtained over the range 50-250 Hz. The choice of driver units was influenced by their availability, and although the sealed rear of the SP 25 is advantageous, this unit has the limitation of a resonant frequency of 400 Hz. Below this frequency the acoustic output of the driver falls off at 12 dB/octave and this seriously influences the behaviour of the absorber. Despite this limitation, however, useful results were obtained. Ideally the resonant frequency should be well below the lowest frequency to be attenuated, but for many applications this would involve the construction of special loudspeakers. The unidirectional microphone consists of two frames each carrying four crystal microphone inserts and mounted on a trolley as shown in Figure 6. A source follower circuit is mounted on each microphone insert to minimise phase changes and hum, and the outputs from each source follower on a ring are summed and amplified before being fed to the “microphone unit” which houses the delay and inverting circuits necessary for unidirectional operation. Another delay unit ensures unidirectional operation of the secondary sources and a third delay is connected between the microphone unit and the secondary sources to maintain the correct time relationship between the detection of the signal and the operation of the secondary sources.
262
J. H. B. POOLE AND H. G. LEVENTHALL
The delay units each consist of a number of simple R-C phase shift networks. Each network is intended to give a small phase shift, which, provided the frequency is low enough, is directly proportional to the frequency. This is equivalent to a constant delay, and by cascading a suitable number of elements any delay can be achieved. This simple circuit enabled a delay of 5 ms constant to about 400 Hz, to be achieved with only 11 stages of phase-shifter and two stages of low-pass filter. This filter was added to the “long” delay unit to remove feedback round the system at high frequencies. Apart from this filter and the 12 dB/octave fall which occurred in the driver unit output, no attempt was made to shape the amplitude-frequency response of the system.
7. RESULTS Two types of detector for the system were used, The first was a dynamic cardioid microphone (AKG D190) and the second consisted of the arrangement of ring microphones, described earlier. All the results were taken without airflow. Further work is required to investigate how airff ow affects the behaviour of the system.
J_ (cl I 140
I.3
180
200
I
I
220
240
Frequency
I
260
I
280
I
300
/
320
(Hz)
Figure 7. Attenuation plotted against frequency for the tirst system, adjusted for maximum attenuation at (a) 162 Hz, (b) 142 Hz and (c) 200 Hz. 0, Experimental results; x, predicted values (on the assumption that the driver unit response falls at 12 dB/octave); A, predicted values (on the assumption that the driver unit response is flat).
The operational frequency range using the dynamic microphone is shown in Figures 7(a), (b) and (c). The readings were obtained by adjusting the “long” delay unit, or the microphone position, and the gains of the amplifiers feeding the secondary sources, until the level of a single frequency tone emitted by the primary source loudspeaker was a minimum as detected by the monitoring microphone. The value of A obtained at that frequency was the difference
ACTIVE ATTENUATION
IN DUCTS
263
in level (in dB’s) indicated on the frequency analyser between the condition with the secondary sources switched on, and with the sources switched off. A was then measured at other frequencies without further adjustment of the system (with the exception of the frequency analyser) to build up the remainder of each curve. This method of setting up the system eliminates the phase and amplitude errors inherent in equation (2) only for the frequency at which the initial adjustments are made. With the simple system described in this paper amplitude and phase errors will seriously af‘fect performance at all other frequencies. The values of A expected from this system were calculated by using equations (2) and (3) and are also plotted on the graphs. For these calculations it was assumed that the output from the driver units fell at 12 dB/octave with decreasing frequency but no account was taken of any phase shifts accompanying this. Also shown on Figure 7(a) are points calculated on the assumption that the drivers had a flat frequency response. The fact that the experimental results were “better” than the predicted ones was attributed to the phase shifts in the driver units to some extent compensating for the phase error given by equation (2). It was encouraging to note that there was no acoustic feedback between the secondary source array and the microphone unless the controls of the amplifiers feeding the two source rings were set at very different relative gains. (When an omnidirectional microphone was used in place of the cardioid the system was on the verge of feedback, but even then 12 dB of attenuation could be obtained at 160 Hz before the system became unstable.) This suggested that the secondary source array was suitably directional, which was confirmed by moving the monitoring microphone along the duct into zone 1. The variation of sound level in the duct with distance is shown in Figure 8. The difference between the maximum and minimum levels of the standing wave upstream of the attenuation is 4.1 dB, which corresponds to a reflection coefficient of 0.23. (A single secondary source gave a value of 30 dB at the same frequency.) The fraction of the sound energy incident on the secondary sources that is reflected back upstream is therefore (0*23)2 or about 5 %. Alternatively one could say that the secondary source array has a front-to-back ratio for its radiation of about 13 dB.
210 220
240
260
280
300
320
Arbifrorydistance scale (cm) Figure 8. Variation of sound level in duct with distance. Conditions as for Figure 7(a) at a frequency of 162 Hz.
The experiment was repeated but with the ring type unidirectional microphone described in section 6 being used. This had a ring spacing of 0.52 metres which was identical to that of the source rings. Two curves are shown, in Figures 9(a) and (b), which were plotted in the same way as Figures 7(a) and (b). The values measured by using the AKG cardioid microphone are also
264
J. H. B. POOLE AND H. G. LEVENTHALL
15
1
(b)
Tlill__L_lkhTTk-
120
100
Frequency
(Hz)
Figure 9. Attenuation plotted against frequency for the second system (with the 2-ring microphone). Maximum attenuation at (a) 162 Hz and (b) 142 Hz. 0, Experimental results; x, predicted values (on the assumption that the driver unit response falls at 12 dB/octave); ?? , experimental values with first system.
plotted for comparison. Although the spaced ring microphones will introduce further amplitude and phase errors according crystal microphones have a much smaller phase variation with could explain why the results with either type of microphone facilities at the authors’ disposal it was not possible to measure phase behaviour of the transducers.)
used in this simple system to equation (2), the fact that frequency than dynamic ones were very similar. (With the accurately the amplitude and
TABLE 2
Attenuation
achieved with each system for d@erent bandwidths of noise Noise bandwidth
Centre frequency
142 180 243
142 180 243
(Hz)
(Hz)
A
/ 0
10
30
loo‘
Attenuation (dB) for first system (dynamic cardioid microphone) 24 19 17 7 25 33
16 25
13 21
9 16
Attenuation (dB) for second system (two spaced microphone rings) 23 15 14 7 32 30
16 25
13 21
10 11
The results so far discussed illustrate system performance with pure tones, but since ductborne noise is in practice broadband, tests were also carried out with band-limited noise. The attenuation achieved with each system adjusted to give maximum attenuation at each of three frequencies is tabulated in Table 2, for three different bandwidths. A comparison of the relative results with Figures 7(b) or 9(b) shows the expected correlation. It can be seen that the attenuation in Table 2 with a noise bandwidth of 0 Hz is somewhat less than that expected from the graphs. The reason for this is that the frequency analyser was tuned to the frequency exercise. of interest while plotting the graphs, but set to “linear ” for the noise measurement The apparent reduction in attenuation is due to harmonic distortion in the loudspeakers and external noise being detected by the sampling microphone.
265
ACTIVEATTENUATIONIN DUCTS TABLE 3 Maximum attenuation achieved at various frequencies with second system Frequency (Hz) Attenuation (dB)
110 43
160 50
210 50
260 48
300 20
It will be noticed from the graphs of attenuation versus frequency that at the frequency at which the system was adjusted for maximum attenuation the loss is considerable, though not infinite. The maximum attenuation achieved at various spot frequencies when using the spaced microphone arrays is given in Table 3. At frequencies below 110 Hz the driver units tended to distort and at 300 Hz the attenuation was limited by acoustic feedback. This feedback was not unexpected since at 300 Hz the combined response of the microphone and secondary sources requires a very high gain in the source amplifiers to give the necessary output. The limit to the attenuation could be attributed to noise or drift in the electronics, or to signals reaching the monitoring microphone from the primary or secondary source by paths other than down the duct, and thus giving rise to a false reading. The most likely cause was the first, since the meter reading on the frequency analyser varied with time after setting up. If this drift were eliminated then the second possibility would require consideration. 8. A PRACTICAL
SYSTEM
The results presented do not suggest that the system, as built, is a suitable replacement for a passive silencer. They do, however, demonstrate a potential which, if pursued further, could result in a practical system. The principal failing of the experimental system is its restricted frequency range. Preliminary experiments on the effects of using a frequency response shaping circuit similar to that described in section 5 of reference [4] have yielded a very useful increase in the frequency range, and it is hoped that the addition of a third ring of sources and microphones will improve this still further. A further refinement discussed by Swinbanks is the incorporation of some form of “tidyup” circuit associated with a microphone placed downstream of the secondary sources. At frequencies well below the duct cut-off frequency this would be useful in correcting for drift in the gains of the amplifiers, but corrections to the phase response of the loop would be more difficult to apply. Again this is a problem for the electronics engineer and its solution has not yet been attempted. Two areas that do require acoustical consideration are the design of suitable transducers and the effect of temperature and flow on the performance of a system. Although the experimental system was designed to operate around 150 Hz the real usefulness of active absorbers is at even lower frequencies where passive devices become impossibly bulky. The necessity for the frequency of resonance of the secondary source driver units to be below the lowest frequency of operation of the system implies that special units will have to be designed. The problem is made slightly more difficult by the need to enclose the rear of the drivers to prevent radiation of sound from there. The design of suitable units should not present insuperable difficulites provided a low conversion efficiency can be compensated for with powerful amplifiers. The diaphragms both of the loudspeakers and of the microphones will, for some applications, have to be robust and resistant to chemical attack, but again this should present few difficulties. The effective speed of sound in a duct is dependent both on the temperature and on the flow rate, the relationships being, respectively, c1 = c,a, and c1 = c,(l + M), where c1 is the effective speed at the temperature Tl or flow rate of math number M, and ce is the speed of sound at temperature TO or under conditions of zero flow. Any change in the speed of
266
J. H. B. POOLE AND H. G. LEVENTHALL
sound will require an adjustment to the delays used in the system. The effect of both these parameters can be minimised by keeping the microphone secondary source separation short, but for high absorption systems some form of compensation will be necessary. This could consist of flow and temperature sensors with suitable control circuitry which would be relatively straightforward to implement, or else compensation could be applied as part of an overall “tidy-up” circuit, which would be more difficult to realise. 9. PRACTICAL
APPLICATIONS
In view of the relatively poor performance of the experimental system and the probable high cost of incorporating some of the developments mentioned, the practical advantage of active absorbers over their passive counterparts is not apparent at first sight. There are, however, several applications where the absence of any static pressure drop or the ability to operate efficiently at low frequencies can amply justify the expense of an active system. In very large air ducting installations, for example, the pressure drop incurred at the silencers necessitates more powerful fans than would otherwise be required to circulate the air. Again, certain industrial installations, notably those involving chimneys, emit sound that is predominantly at one frequency. It might be that even a relatively simple active absorber, similar to the experimental system described here, could be used successfully in this case.
10. CONCLUSION The theory of active absorption is now well developed. The problems in producing practical systems are centred on developing configurations of secondary sources which satisfy the requirements laid down in the theory. These are (1) that there should be no radiation from the secondary sources in the direction of the noise source, and (2) that the radiation from these sources towards the zone to be protected must be an accurate, but inverted, reproduction of the noise to be cancelled. The two-ring source described by Swinbanks satisfies the first of these conditions, but only approximates to the second over a limited frequency range. The results of the experiments so far completed suggest that a two-ring source array can usefully reduce noise that is predominantly at one frequency, and that extension of the frequency range by the addition of a third ring source and development of suitable control circuitry could yield a very versatile unit. Such a unit might be expensive to produce, but the advantages of low pressure drops and high attenuation at low frequencies could compensate for the additional cost of an active system.
REFERENCES 1. M. J. M. JESSELand G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390.
2. 3. 4. 5.
Active sound absorbers in an air duct. G. CANBVETand G.A. MANGIANTE 1974 Acustica 30,4&48. Absorption acoustique active et antibruit a une dimension. E. c&CON1 1973 M.Sc. Dissertation, Chelsea College, University oflondon. Active sound absorption in ducts. M. A. SWINBANKS 1973 Journal of Sound and Vibration 28,411436. The active control of sound propagation in long ducts. P. LUEG 1936 U.S. Patent No. 2 043 416. Process of silencing sound oscillations.
AN EXPERIMENTAL OF ACTIVE
STUDY
ATTENUATION
OF SWINBANKS’ OF SOUND
METHOD
IN DUCTS
J. H. B. POOLE AND H. G. LEVENTHALL Department of Physics, Chelsea College, Pulton Place, London S W6 SPR, England (Received 3 May 1976, and in revisedform 29 July 1976)
The concept of attenuating sound propagating down a duct by means of an antiphase copy of the sound was first put forward forty years ago, but to date no practical system has been produced. The principal problem is to introduce the antiphase signal in such a way that it propagates only in the direction of propagation of the original sound and to ensure that this property can be maintained over a useful frequency range. A suitable arrangement of secondary sound sources has recently been proposed by Swinbanks and this paper describes experiments carried out in an attempt to realise a practical system. A unidirectional array of secondary sources has been successfully constructed around a rectangular duct, using loudspeaker drive units and electronic delays. Sound propagating in the direction of these sources was sampled and a control signal applied to the sources which in turn acted to significantly reduce the amplitude of the sound. Pure tones at frequencies around 150 Hz have been attenuated by more than 50 dB but results with band-limited noise have been less successful. Further work is suggested which should result in a device having significant advantages over conventional splitter silencers, at low frequencies. 1. INTRODUCTION Existing methods of attenuating sound propagating in ducts include absorptive silencers, lined ducts, and plenum chambers. The first two techniques work well for sounds of medium and high frequencies, but they are very inefficient at low frequencies. Plenum chambers designed to work at low frequencies tend to be bulky. The technique of active absorption, in which cancellation is used to reduce sound fields, provides a means of achieving high degrees of attenuation at low frequencies as well as at the higher frequencies. The size and pressure drops are also less than with passive systems. Previous recent work in the field [l-3], has concentrated on attempts to achieve high absorption at discrete frequencies. The present work draws largely on theory developed by Swinbanks [4], and is aimed at achieving a significant degree of absorption over a band of frequencies. The signal which interferes with the unwanted noise is derived directly from the sound field in the duct so that it becomes possible to operate on complex “real life” waveforms.
2. REQUIREMENTS
FOR ACTIVE
ATTENUATION
Figure 1 is a sketch of an idealised active sound absorber for an air duct. Noise propagating down the duct is detected by a microphone which supplies the information necessary for the operation of a system of secondary noise sources. These secondary sources generate a sound pressure field in antiphase with the noise from the original source, effectively silencing the air stream. In an ideal duct, all frequencies below the first cut-off frequency of the duct must propagate as plane waves. Thus it would appear that a single loudspeaker mounted on the duct wall 25-l
258
J. H. B. POOLE AND H. G. LEVENTHALL
~w Figure 1. A simple representation of an active absorber.
would be sufficient to create an accurate copy of the noise signal, the required phase being obtained by suitably delaying the signal from the microphone. Such a system was the subject of a patent in 1936 [5], and a recent experiment carried out by the authors achieved a reduction in the downstream’sound level of a single frequency tone of about 50 dB by this means. (Throughout this paper reference is made to cancellation downstream of the noise source. Similar techniques can be applied to cancellation upstream of a source.) The loudspeaker does not exchange energy with the noise field in the duct, but is analogous to an impedance change and, instead of being absorbed, the noise is effectively reflected back up the duct. A simple theory based on Young’s theory of interference and Huygen’s Principle has been developed by Jesse1 [I]. He has shown that if an arrangement of secondary sources can be found, which in the absence of any incident sound wave can generate a sound wave that propagates only in the downstream direction, then the incident wave will not be reflected, but absorbed by the sources. One suitable arrangement consists of a combination of a monopole and a dipole source. Experiments carried out by Jesse1 and others [l-3], using this combination, have achieved attenuations of up to 70 dB at discrete frequencies. In the Swinbanks arrangement spaced arrays of monopole sources are used to achieve unidirectionality. Both systems require electronic delays and electronic filtering to achieve satisfactory operation over a useful range of frequencies. A weakness shared by both systems is that they only operate on sounds in the plane-wave mode. Thus a practical duct silencing system will need to retain passive absorption methods for the higher order modes (which are relatively easy to attenuate in this way) while using active methods to remove the lowest frequencies which are more difficult to remove by conventional means. 3. SUMMARY
OF SWINBANKS’
WORK
One secondary source arrangement derived by Swinbanks consists of a pair of source “rings” mounted on the duct walls as shown schematically in Figure 2. Each ring consists of four loudspeaker units, one mounted centrally on each face of the (rectangular) duct. If the source strength of the upstream ring at time t is ml(t) and that of the downstream ring is mz(t) then Swinbanks shows that there will be zero output from the combination of the two rings in the upstream direction if q(t)
= -mz(t - b/c&
- M)),
(1)
where b is the ring spacing, c0 is the velocity of sound in still air, and M is the Mach number of the air-flow in the duct. (A positive value of M denotes flow from left to right in Figure 1.) Thus ml must have the same amplitude as mZ but be inverted, and delayed by b/cO(l - M) relative to m2. There will, however, be an output (D) in the downstream direction which will depend on the angular frequency (0) according to the expression D a
sin (0.42) cos w{t - (x - b)/c,( 1 + M) - 2,/2},
(2)
ACTIVEATTENUATION IN DUCTS
259
CL$f?Q ,&--_-_____y
_____,7L-_-___--
Figure 2. The dispositionof secondarysources around a rectangular duct to form a 2-ring array.
n/
277/r, 2%
To
%I
41r/r, PW”,’ 5 4wm m
Angular frequency, w
Figure 3. The downstreamamplitude-frequencyresponseof the 2-ringarray. where z0 = 2b/c,(l - W). This expression is plotted in Figure 3. The shape of this curve is by no means ideal for the purpose of broad-band active absorption, but experiments in which this arrangement has been used have produced some useful results. By adding a further ring source downstream of the other two the useful frequency range can be extended, but since no experiments were carried out with this arrangement it will not be considered here. Details will be found in reference [4]. The reason for the use of ring sources is not given in the theory above, but is considered in detail in the original paper. If all frequencies of interest lie below o,, the lowest cut-off frequency of the duct, then a single loudspeaker unit at each position, x = 0 and x = b, will suffice. Above this frequency, however, a single unit will excite transverse modes in the duct, as well as the plane wave mode, even if the primary disturbance is entirely in the plane wave mode. By using a ring source, the lowest frequency at which transverse modes can be excited becomes 2.8 o, for a square duct. For a circular duct, three equally spaced sources are used, and the factor is 2.1 0,. 4. OTHER SYSTEM REQUIREMENTS Apart from the secondary source array itself, the other necessary components of the system shown in Figure 1 are the microphone and the electronics necessary to convert the microphone signal into a suitable form for driving the secondary sources. The requirements for the processing section are to delay the microphone signal to allow the detected wavefront time to progress to the cancellation sources, to realise equation (I), to shape the response of the input to the To secondary sources
Delay Microphone
TI
Figure 4. Block diagram of a practical active absorber in which a 2-ring secondary source array is used.
J. H. B. POOLE AND H. G. LEVENTHALL
260
secondary sources to compensate in part for the amplitude-frequency response of the source array, and to amplify the processed signal to the required level. These functions are shown in block diagram form in Figure 4. The requirements for the microphone are that there should be no phase shift between the pressure input and the electrical output, or alternatively that any phase shift should be directly proportional to frequency and thus be equivalent to a simple delay. With a perfectly unidirectional secondary source array there should, in theory, be no problem of feedback between the sources and the microphone. In practice, however, there is a danger of feedback from a source array that is not fully directional and also from reflections of imperfectly cancelled noise from further downstream. It is thus desirable to use a microphone that is only sensitive to sound travelling downstream. Such a microphone could be constructed in a fashion similar to that for the unidirectional source. This would consist of two (or three) rings of microphones, the output from the downstream ring(s) being inverted and delayed before being added to the output from the upstream ring. This array would have a similar amplitude/frequency response to that of the secondary source array (Figure 3). For further details of a unidirectional microphone the reader is referred to section 5.2 of reference [4]. 5. THE CONCEPT
OF AN ATTENUATION
COEFFICIENT
For total cancellation of the incident sound downstream of the attenuator the secondary source array must produce a perfect antiphase copy of this sound field. In practice there are likely to be small errors in the phase and amplitude of the cancellation signal and the effectiveness of the system under these conditions can be expressed in terms of an “attenuation coefficient”, A, defined by A = lOlog,,Pf/P& where PI is the acoustic pressure of the sound in zone 1 and P3 the pressure in zone 3 of Figure 1. Alternatively A = 2010g~o(l~pl/l~p
+PI)
where pp is the acoustic pressure at a point in zone 3 due to the primary the pressure at the same point due to the secondary sources.
noise source and p”
s Figure 5. Vector diagram for the calculation of the attenuation coefficient. If, for example, pp = A cos~t, then complete cancellation will be achieved if p” = -A cos wt. If one supposes that, instead, p” turns out to be -Bcos(ot + 6rp), then the value of A can be calculated by vector addition, as shown in Figure 5, to be Ipp + @I = z/W
- B cos 6~)~ +
(B sin &)‘I
=
Az/l 1 -
2(R/A) cos &p + B2/A21
TABLE 1
Theoretical
attenuation coefficient (dB) achievable various conditions of imperfection
under
B/A
i&p = 0”
sq5= 1”
l&$=3”
sfj = 10’
1 1.005 1.05 1.1
4: 26 20
35 33.7 25.7 19.8
25.5 25-4 22.6 18.6
15.2 15.1 14.2 13.6
ACTIVE ATTENUATION
(time variation
being ignored).
Since lppl = A, the attenuation
A = -lOlog,O(jl
261
IN DUCTS
- 2(B/A)cos@
coefficient
becomes
+ B2/A2\}.
(3)
It should be noted that any phase difference will worsen the attenuation coefficient. Thus i$ is not possible to compensate for changes in B/A with changes in hp. To illustrate the effect of small errors on the attenuation obtained, a few values of A calculated by using equation (3) are given in Table 1.
6. THE EXPERIMENTAL
SYSTEM
The experimental arrangement is sketched in Figure 6. The primary source is represented by a loudspeaker at one end of the duct, the other end of which is left open. The outer sections are each 2.5 metres long and made of 25 mm blockboard, while the central section is 0.75 metres long, of similar construction but with a “perspex” front. The microphones may be moved at will in the duct by means of an arrangement of strings and pulleys. Au--------------I ’
Secondary
sources
(Not to scale)
Figure 6. A sketch of the experimental apparatus.
The secondary source rings each consist of four Grampian SP 25 horn driver units, one mounted centrally on each face of the duct as shown, and connected in series. The ring spacing was chosen to give a “centre frequency” (w,, on Figure 3) of about 150 Hz, the actual dimensions arrived at giving 162 Hz. It was hoped that with this centre frequency useful results would be obtained over the range 50-250 Hz. The choice of driver units was influenced by their availability, and although the sealed rear of the SP 25 is advantageous, this unit has the limitation of a resonant frequency of 400 Hz. Below this frequency the acoustic output of the driver falls off at 12 dB/octave and this seriously influences the behaviour of the absorber. Despite this limitation, however, useful results were obtained. Ideally the resonant frequency should be well below the lowest frequency to be attenuated, but for many applications this would involve the construction of special loudspeakers. The unidirectional microphone consists of two frames each carrying four crystal microphone inserts and mounted on a trolley as shown in Figure 6. A source follower circuit is mounted on each microphone insert to minimise phase changes and hum, and the outputs from each source follower on a ring are summed and amplified before being fed to the “microphone unit” which houses the delay and inverting circuits necessary for unidirectional operation. Another delay unit ensures unidirectional operation of the secondary sources and a third delay is connected between the microphone unit and the secondary sources to maintain the correct time relationship between the detection of the signal and the operation of the secondary sources.
262
J. H. B. POOLE AND H. G. LEVENTHALL
The delay units each consist of a number of simple R-C phase shift networks. Each network is intended to give a small phase shift, which, provided the frequency is low enough, is directly proportional to the frequency. This is equivalent to a constant delay, and by cascading a suitable number of elements any delay can be achieved. This simple circuit enabled a delay of 5 ms constant to about 400 Hz, to be achieved with only 11 stages of phase-shifter and two stages of low-pass filter. This filter was added to the “long” delay unit to remove feedback round the system at high frequencies. Apart from this filter and the 12 dB/octave fall which occurred in the driver unit output, no attempt was made to shape the amplitude-frequency response of the system.
7. RESULTS Two types of detector for the system were used, The first was a dynamic cardioid microphone (AKG D190) and the second consisted of the arrangement of ring microphones, described earlier. All the results were taken without airflow. Further work is required to investigate how airff ow affects the behaviour of the system.
J_ (cl I 140
I.3
180
200
I
I
220
240
Frequency
I
260
I
280
I
300
/
320
(Hz)
Figure 7. Attenuation plotted against frequency for the tirst system, adjusted for maximum attenuation at (a) 162 Hz, (b) 142 Hz and (c) 200 Hz. 0, Experimental results; x, predicted values (on the assumption that the driver unit response falls at 12 dB/octave); A, predicted values (on the assumption that the driver unit response is flat).
The operational frequency range using the dynamic microphone is shown in Figures 7(a), (b) and (c). The readings were obtained by adjusting the “long” delay unit, or the microphone position, and the gains of the amplifiers feeding the secondary sources, until the level of a single frequency tone emitted by the primary source loudspeaker was a minimum as detected by the monitoring microphone. The value of A obtained at that frequency was the difference
ACTIVE ATTENUATION
IN DUCTS
263
in level (in dB’s) indicated on the frequency analyser between the condition with the secondary sources switched on, and with the sources switched off. A was then measured at other frequencies without further adjustment of the system (with the exception of the frequency analyser) to build up the remainder of each curve. This method of setting up the system eliminates the phase and amplitude errors inherent in equation (2) only for the frequency at which the initial adjustments are made. With the simple system described in this paper amplitude and phase errors will seriously af‘fect performance at all other frequencies. The values of A expected from this system were calculated by using equations (2) and (3) and are also plotted on the graphs. For these calculations it was assumed that the output from the driver units fell at 12 dB/octave with decreasing frequency but no account was taken of any phase shifts accompanying this. Also shown on Figure 7(a) are points calculated on the assumption that the drivers had a flat frequency response. The fact that the experimental results were “better” than the predicted ones was attributed to the phase shifts in the driver units to some extent compensating for the phase error given by equation (2). It was encouraging to note that there was no acoustic feedback between the secondary source array and the microphone unless the controls of the amplifiers feeding the two source rings were set at very different relative gains. (When an omnidirectional microphone was used in place of the cardioid the system was on the verge of feedback, but even then 12 dB of attenuation could be obtained at 160 Hz before the system became unstable.) This suggested that the secondary source array was suitably directional, which was confirmed by moving the monitoring microphone along the duct into zone 1. The variation of sound level in the duct with distance is shown in Figure 8. The difference between the maximum and minimum levels of the standing wave upstream of the attenuation is 4.1 dB, which corresponds to a reflection coefficient of 0.23. (A single secondary source gave a value of 30 dB at the same frequency.) The fraction of the sound energy incident on the secondary sources that is reflected back upstream is therefore (0*23)2 or about 5 %. Alternatively one could say that the secondary source array has a front-to-back ratio for its radiation of about 13 dB.
210 220
240
260
280
300
320
Arbifrorydistance scale (cm) Figure 8. Variation of sound level in duct with distance. Conditions as for Figure 7(a) at a frequency of 162 Hz.
The experiment was repeated but with the ring type unidirectional microphone described in section 6 being used. This had a ring spacing of 0.52 metres which was identical to that of the source rings. Two curves are shown, in Figures 9(a) and (b), which were plotted in the same way as Figures 7(a) and (b). The values measured by using the AKG cardioid microphone are also
264
J. H. B. POOLE AND H. G. LEVENTHALL
15
1
(b)
Tlill__L_lkhTTk-
120
100
Frequency
(Hz)
Figure 9. Attenuation plotted against frequency for the second system (with the 2-ring microphone). Maximum attenuation at (a) 162 Hz and (b) 142 Hz. 0, Experimental results; x, predicted values (on the assumption that the driver unit response falls at 12 dB/octave); ?? , experimental values with first system.
plotted for comparison. Although the spaced ring microphones will introduce further amplitude and phase errors according crystal microphones have a much smaller phase variation with could explain why the results with either type of microphone facilities at the authors’ disposal it was not possible to measure phase behaviour of the transducers.)
used in this simple system to equation (2), the fact that frequency than dynamic ones were very similar. (With the accurately the amplitude and
TABLE 2
Attenuation
achieved with each system for d@erent bandwidths of noise Noise bandwidth
Centre frequency
142 180 243
142 180 243
(Hz)
(Hz)
A
/ 0
10
30
loo‘
Attenuation (dB) for first system (dynamic cardioid microphone) 24 19 17 7 25 33
16 25
13 21
9 16
Attenuation (dB) for second system (two spaced microphone rings) 23 15 14 7 32 30
16 25
13 21
10 11
The results so far discussed illustrate system performance with pure tones, but since ductborne noise is in practice broadband, tests were also carried out with band-limited noise. The attenuation achieved with each system adjusted to give maximum attenuation at each of three frequencies is tabulated in Table 2, for three different bandwidths. A comparison of the relative results with Figures 7(b) or 9(b) shows the expected correlation. It can be seen that the attenuation in Table 2 with a noise bandwidth of 0 Hz is somewhat less than that expected from the graphs. The reason for this is that the frequency analyser was tuned to the frequency exercise. of interest while plotting the graphs, but set to “linear ” for the noise measurement The apparent reduction in attenuation is due to harmonic distortion in the loudspeakers and external noise being detected by the sampling microphone.
265
ACTIVEATTENUATIONIN DUCTS TABLE 3 Maximum attenuation achieved at various frequencies with second system Frequency (Hz) Attenuation (dB)
110 43
160 50
210 50
260 48
300 20
It will be noticed from the graphs of attenuation versus frequency that at the frequency at which the system was adjusted for maximum attenuation the loss is considerable, though not infinite. The maximum attenuation achieved at various spot frequencies when using the spaced microphone arrays is given in Table 3. At frequencies below 110 Hz the driver units tended to distort and at 300 Hz the attenuation was limited by acoustic feedback. This feedback was not unexpected since at 300 Hz the combined response of the microphone and secondary sources requires a very high gain in the source amplifiers to give the necessary output. The limit to the attenuation could be attributed to noise or drift in the electronics, or to signals reaching the monitoring microphone from the primary or secondary source by paths other than down the duct, and thus giving rise to a false reading. The most likely cause was the first, since the meter reading on the frequency analyser varied with time after setting up. If this drift were eliminated then the second possibility would require consideration. 8. A PRACTICAL
SYSTEM
The results presented do not suggest that the system, as built, is a suitable replacement for a passive silencer. They do, however, demonstrate a potential which, if pursued further, could result in a practical system. The principal failing of the experimental system is its restricted frequency range. Preliminary experiments on the effects of using a frequency response shaping circuit similar to that described in section 5 of reference [4] have yielded a very useful increase in the frequency range, and it is hoped that the addition of a third ring of sources and microphones will improve this still further. A further refinement discussed by Swinbanks is the incorporation of some form of “tidyup” circuit associated with a microphone placed downstream of the secondary sources. At frequencies well below the duct cut-off frequency this would be useful in correcting for drift in the gains of the amplifiers, but corrections to the phase response of the loop would be more difficult to apply. Again this is a problem for the electronics engineer and its solution has not yet been attempted. Two areas that do require acoustical consideration are the design of suitable transducers and the effect of temperature and flow on the performance of a system. Although the experimental system was designed to operate around 150 Hz the real usefulness of active absorbers is at even lower frequencies where passive devices become impossibly bulky. The necessity for the frequency of resonance of the secondary source driver units to be below the lowest frequency of operation of the system implies that special units will have to be designed. The problem is made slightly more difficult by the need to enclose the rear of the drivers to prevent radiation of sound from there. The design of suitable units should not present insuperable difficulites provided a low conversion efficiency can be compensated for with powerful amplifiers. The diaphragms both of the loudspeakers and of the microphones will, for some applications, have to be robust and resistant to chemical attack, but again this should present few difficulties. The effective speed of sound in a duct is dependent both on the temperature and on the flow rate, the relationships being, respectively, c1 = c,a, and c1 = c,(l + M), where c1 is the effective speed at the temperature Tl or flow rate of math number M, and ce is the speed of sound at temperature TO or under conditions of zero flow. Any change in the speed of
266
J. H. B. POOLE AND H. G. LEVENTHALL
sound will require an adjustment to the delays used in the system. The effect of both these parameters can be minimised by keeping the microphone secondary source separation short, but for high absorption systems some form of compensation will be necessary. This could consist of flow and temperature sensors with suitable control circuitry which would be relatively straightforward to implement, or else compensation could be applied as part of an overall “tidy-up” circuit, which would be more difficult to realise. 9. PRACTICAL
APPLICATIONS
In view of the relatively poor performance of the experimental system and the probable high cost of incorporating some of the developments mentioned, the practical advantage of active absorbers over their passive counterparts is not apparent at first sight. There are, however, several applications where the absence of any static pressure drop or the ability to operate efficiently at low frequencies can amply justify the expense of an active system. In very large air ducting installations, for example, the pressure drop incurred at the silencers necessitates more powerful fans than would otherwise be required to circulate the air. Again, certain industrial installations, notably those involving chimneys, emit sound that is predominantly at one frequency. It might be that even a relatively simple active absorber, similar to the experimental system described here, could be used successfully in this case.
10. CONCLUSION The theory of active absorption is now well developed. The problems in producing practical systems are centred on developing configurations of secondary sources which satisfy the requirements laid down in the theory. These are (1) that there should be no radiation from the secondary sources in the direction of the noise source, and (2) that the radiation from these sources towards the zone to be protected must be an accurate, but inverted, reproduction of the noise to be cancelled. The two-ring source described by Swinbanks satisfies the first of these conditions, but only approximates to the second over a limited frequency range. The results of the experiments so far completed suggest that a two-ring source array can usefully reduce noise that is predominantly at one frequency, and that extension of the frequency range by the addition of a third ring source and development of suitable control circuitry could yield a very versatile unit. Such a unit might be expensive to produce, but the advantages of low pressure drops and high attenuation at low frequencies could compensate for the additional cost of an active system.
REFERENCES 1. M. J. M. JESSELand G. A. MANGIANTE 1972 Journal of Sound and Vibration 23, 383-390.
2. 3. 4. 5.
Active sound absorbers in an air duct. G. CANBVETand G.A. MANGIANTE 1974 Acustica 30,4&48. Absorption acoustique active et antibruit a une dimension. E. c&CON1 1973 M.Sc. Dissertation, Chelsea College, University oflondon. Active sound absorption in ducts. M. A. SWINBANKS 1973 Journal of Sound and Vibration 28,411436. The active control of sound propagation in long ducts. P. LUEG 1936 U.S. Patent No. 2 043 416. Process of silencing sound oscillations.