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 Elliott Sound Products The Subwoofer Conundrum 

Copyright © 2004 - Rod Elliott (ESP)
Page Created 01 Mar 2004

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Contents
1 - Introduction

Conundrum (noun): A paradoxical, insoluble, or difficult problem. A dilemma. Well, that just about sums it up for subwoofers, doesn't it? Of all the add-ons that can be applied to a system, the sub often gets the short straw - they are routinely shoved into a corner, or behind the couch, or even made into coffee tables. The problem is that none of these treatments will work properly unless you have some idea of the physics behind it. The vast number of compromises needed only makes things harder.

The use of subwoofers in any audio system that is intended mainly for music is often the cause of much soul-searching, not to mention web searching, trying to find out what will work and what won't. Home theatre is easier, since the primary requirement is for satisfyingly deep rumbles at the appropriate times, and to give the sense of "depth" to the soundtrack, and accuracy is not required to the same degree. Yes, I know that many consider film soundtracks to warrant accuracy too, but most listeners/ viewers don't care that much.

In the case of a home theatre system, there is usually no concept of "speed", since the sounds do not usually have the tight connection found in music, and a few milliseconds here or there is of little consequence. Not so if the system will be used for music as well, since now there are distinct notes that should be reproduced with as little time delay as possible. If this is not the case, the bass becomes blurred and indistinct, and has no proper connection to the music overlaying the lowest notes.

People often describe a bass reproducer as "slow" or "fast", but in fact neither term is applicable. A 40Hz bass fundamental cannot be fast or slow - it is simply a 40Hz (transient) tone, and our hearing is depressingly bad at even hearing such frequencies until they have been present for several cycles.

So, do the terms actually mean anything, or is this more fluff and bluster from the marketing departments of the larger manufacturers?


2 - Timing is Everything

When used in any advertising, you have basically fluff and bluster, and little else. Bass cannot be slow or fast, but it can be reproduced and heard when you are supposed to hear it, or it can be out of time with the rest of the music. The timing in itself is usually not the problem, but there are many things afoot that make a huge difference. These (not surprisingly) are the subject of this article.

Consider the setup shown in Figure 1. The sub is off to one side, with the listener placed directly in front of the main speakers. With spacings as shown, the distance from the main speakers to the listener is 2.23 metres to each box. Subwoofer to listener is 3.6 metres, a path length difference of 1.37 metres. This is not an uncommon setup, and as is often the case, the WAF (Wife Acceptance Factor) must be considered in the final placement. It is also non-sensible to have a subwoofer in the middle of the floor for people to trip over


Figure 1 - Typical Listening Room Setup

At higher frequencies, a path difference of 1.4 metres (near enough) would be completely unacceptable, but it can also be unacceptable for bass, too. It depends on the crossover frequency. The following table shows the important frequencies and wavelengths at various frequencies.

Frequency1/4 Wave (-3dB)1/2 Wave (cancels!) 3/4 Wave (-3dB)Wavelength
120Hz0.7181.442.162.875
100Hz0.86251.7252.8753.45
80Hz1.0782.1563.234.3125
60Hz1.43752.8754.31255.75
40Hz2.1564.316.478.625
Table 1 - Frequency Vs. Wavelength (in metres)

For the purposes of this exercise, I have no choice but to assume a perfect (anechoic) listening space, devoid of reflections and standing waves. Unfortunately, these always exist, and rarely improve matters, but may mask some of the effects that will be described. No matter, since the basic principles stand and should be considered - the effects of standing waves (in particular) are a separate topic altogether.

Over the entire crossover range, if the difference between the listener and main speakers vs. the distance from listener to sub is 1/2 wavelength, then the sub's output will partially cancel the output from the main speakers. A phase switch is often included to fix this, and this is the most obvious of all problems. In the case shown in Figure 1, 120Hz is a crossover frequency to be avoided at all costs, since there is almost a complete cancellation due to relative path lengths. Remember that the crossover range should be considered to be a full octave with typical crossovers, and much higher slopes are needed to minimise this. 12dB/Octave is an absolute minimum requirement, and 24dB/Octave should be considered the standard.

But, why can't we just operate the phase switch and make it right again? Well, we can, but then there are other problems. What of the path length between the sub and the speakers themselves, and what of the frequencies either side of crossover? Since these are all different, there will be interactions at the main speakers, at the sub, and at various places throughout the room. Note that in an ideal case, 1/4 or 3/4 wavelengths will cause a dip of 3dB, since they have a 90° and 270° degree phase angle respectively. If at all possible, the relative path lengths should be no more than 1/4 wavelength for at least 1/2 octave above crossover - this effectively rules out the layout shown in Figure 1, but in reality it will work provided the crossover frequency is no higher than 60Hz or so.

Every multiple of 1/2 wavelength throughout the room will cause a dip in response, which may be partial or total (no signal at all), and this will be effective to a greater or lesser degree for the whole range of frequencies across the crossover region. With a setup such as this, the crossover frequency should be no higher than 60Hz, or the subwoofer must be moved to a more central location, closer to the main speakers.

At any crossover frequency higher than around 60Hz, standing waves, which are influenced by the size of the room, the wall, floor and ceiling materials, (large) soft furnishings, etc., will confuse an already confused bass image, and may well give rise to the impression that it is 'slow'. Bass signals around the crossover frequency will be of variable amplitude, and may change from one note to the next. Despite claims to the contrary in some areas, you will also probably find that you can tell where the bass is coming from as well - there will be enough higher frequency energy from a 120Hz crossover to enable you to localise the bass as being well off-centre.


3 - Optimum Crossover Frequency

Generally, a sub should be crossed over at the lowest practicable frequency. Based on the information in Table 1, you may need to re-think the idea of just shoving it behind a lounge chair, based on the crossover frequency and the relative distances from main speakers (both of them!) to the listener and the sub itself.

An ideal solution would be to use two subs, each located near to the speakers. Unfortunately, cost, size and even the ability to place them optimally often (usually) rules out this approach. The next best thing is to try to arrange it so that the path length between the sound sources and the listener are within at least 1/4 wavelength at the crossover frequency. Unfortunately, this will often mean a location that is already occupied by other furnishings, the TV or even the cat

The limitations on crossover frequency are alleviated somewhat by positioning, but the optimum position for the sub to minimise phase and timing issues may well turn out to the worst possible location for standing waves and good bass at the listening position. As shown in Figure 2, by re-positioning the sub so that it is between the two main speakers will allow a crossover frequency that is much higher than the previous example, but the room shape and size may make this position unacceptable for standing waves.


Figure 2 - Alternative Subwoofer Positioning

If this works well for bass at the listening positioning, then based on timing errors, the maximum phase shift is negligible, and well under 90° (path length difference is reduced to 230mm) even with a 100Hz crossover frequency. If this cannot be achieved, then the alternatives are few.

Adjustable phase controls usually do more harm than good, since the final phase relationships are something of a lottery. You might win, but the odds are stacked against you. Digital delay will work too, but it cannot correct off-centre positioning where the distances from each box to the listener and between each other are all different. One of the few tools at your disposal (for a sensible price at least) is an equaliser (for example the Sub-Woofer Equaliser shown in the projects section. It is not a panacea, but can help if things aren't quite right, and can correct many problems that cannot be fixed any other way. The other thing (of course) is to optimise the crossover frequency, getting it as low as you can - the limiting factor is often the main speakers (especially if smaller bookshelf types), but most speakers will get down to 70Hz, and many a lot deeper.

There is always an inevitable tradeoff between the minimum frequency of the main speakers and the likelihood of intermodulation distortion caused by excessive cone excursions - this is relieved somewhat by using a 3-way main system, but this is not always practical. Additional phase anomalies are introduced as any loudspeaker approaches resonance, again, there is little you can do about it, but careful placement is a better solution than variable phase controls in most cases.


4 - Cone Area Vs. Displacement

A common misconception is that cone area can be reduced if displacement (Xmax) is increased. While it is true up to a point, it is only possible to increase Xmax up to a few millimetres of linear travel without a severe sacrifice in loudspeaker driver efficiency. Consider the two options shown in Figure 3 - the voicecoil may be overhung or underhung, but one either loses efficiency by having only a part of the coil in the gap, or by having much of the magnetic field bypassing the voicecoil altogether. Low efficiency (less than 90dB/m/W) requires more amplifier power and more heat. Remember that each 3dB fall in efficiency requires double the amplifier power for the same SPL.

Also, many manufacturers rate efficiency at 2.83V instead of 1W. At 8 ohms, the figures are the same, but for a 4 ohm driver, the efficiency is artificially inflated by 3dB, and you will need twice as much power as you may have thought.


Figure 3 - Voicecoil Geometries for Large Xmax

In each case, linear Xmax is defined as the distance the voicecoil can travel, while remaining within the magnetic field. Xmax is often stated as the maximum cone travel before the suspension prevents further movement (or the coil former hits the rear polepiece, which will damage the former). IMO this is very misleading, and while the cone may well be capable of travel to the distances (Xmax) claimed, the distortion becomes unacceptably high. Distortion can easily exceed 20% in many cases.

This is only a part of the problem however. For the sake of easy explanation, imagine a cone 25mm in diameter, but with a linear Xmax of 350mm. It would reproduce little or no bass at all, despite its massive excursion. At the other end of the scale, a 350mm diameter loudspeaker cone with an Xmax of 25mm will be an efficient reproducer of very low frequencies. The problem with generating bass using excursion to 'replace' diameter is the radiation impedance of the cone area. Cone loudspeakers are inefficient at the best of times, not (only) because the coil and magnetic circuits are not always optimised, but because there is a huge mismatch between the impedance of the cone and that of the air which must carry the soundwaves. This is but one of many compromises that must be made in any loudspeaker driver design.

An ideal bass radiator has a cone that is large, very light (or at least a lot lighter than most of the current crop of subwoofers), is still strong, and has a low resonant frequency. Unfortunately, this is very difficult to achieve unless the suspension is made very "floppy" indeed, and this will cause problems since it is not possible for such a suspension to keep the voicecoil properly centred when the speaker is driven hard. A low resonant frequency with a light cone also means that the driver will be very sensitive to the enclosure size (large Vas), and will need a big box to work well. This all goes against the current trends (and the need for the sub to be "acceptably invisible" in the listening room).

Horn loudspeakers minimise the impedance mismatch between diaphragm and the air, by acting as an acoustical transformer. The horn flare provides a controlled expanding wavefront to transform relatively small diaphragm movement into a large air movement, and with careful design efficiencies of well over 100dB/W/m are easily achieved. Unfortunately, bass is again a problem, since the circumference of the mouth should be equal to the wavelength of the lowest frequency to be reproduced. At 20Hz, this means a circumference of over 17 metres, and for a square horn (less than ideal, but smaller than circular) that means 4.3 metres per side! That is very large indeed, and the length hasn't even been considered yet. To avoid impedance mismatch (which causes "ripples" - peaks and dips in the frequency response), a horn should have a minimum length of 1/4 wavelength , and preferably more. This is not a practical solution for most listening rooms.

Radiation impedance is a complex area of acoustics, and I'm not going to even try to explain the maths involved. Suffice to say that small cones (in small boxes) will be incapable of reproducing spectacular bass, regardless of the claimed Xmax of the driver. As an example (and please note that this is a very rough estimate only) the following table shows the expected minimum frequency of various sized drivers in a sealed enclosure having a small baffle, and radiating into ½ space. Vented boxes work differently, but the vent area still limits the low frequency performance. A very large vent (needed for good performance and low noise at extra low frequencies) in an enclosure cannot be effectively driven with a small loudspeaker cone area.

Driver DiameterMinimum Frequency (-3dB)
200mm (8")40 Hz
250mm (10")32 Hz
300mm (12")27 Hz
380mm (15")21 Hz
450mm (18")18 Hz
Table 2 - Diameter Vs. Minimum Frequency (½ Space)

The above figures should be taken as a guide only, and are based on wave propagation (as opposed to pressure mode - see below). The figures are independent of Xmax!  When the subwoofer is located at a room boundary (e.g. close to the wall and sitting on the floor) there is an increase of low frequency energy, and a driver will reproduce to a lower frequency than indicated, but with poorer frequency linearity. Corner location gives a further boost, but at the expense of further frequency non-linearities. Vented (or passive radiator) boxes perform differently, and it is not possible to cover all the permutations here.

It has been claimed (and it is true up to a point) that distortion of extreme bass is not a problem, since it is masked by the main system, and may be rendered inaudible. While this is fine in theory, remember that the distortion will be primarily 3rd and 5th harmonics. If a 50Hz signal has significant distortion because Xmax has been exceeded, then you will hear 150Hz and 250Hz harmonics. Further, these will allow precise location of the sub, since 250Hz is easily high enough to allow our ears to localise the sound source.

The same thing happens with vented subs - if vent noise becomes audible during low frequency programme transients, then you will again be able to pinpoint the sub's location quite easily. This detracts greatly from the listening experience. In all cases, a hi-fi system should be as transparent as possible, and do nothing to draw attention to itself. Only the programme material is important, and it is this - and this only - that you should be listening to.


5 - Soundwave Vs. Pressure Mode

One claim that is very popular is that you cannot reproduce bass in a small space. While this is certainly true of soundwave propagation, it is completely false if pressure mode is excited within the space. In most rooms, there is a transition point between soundwave propagation and pressure mode (sometimes referred to as 'Room Gain'). That low frequencies can be reproduced in small spaces is clearly shown by headphones and car subwoofer systems. Both of these rely on pressure mode, and the low frequency -3dB point is defined by 'leakage' because the space is not perfectly sealed. In a completely sealed environment, bass reproduction is attainable right down to DC - not that this is actually useful. Most subwoofers ultimately rely on pressure mode to obtain the lowest frequencies in typical rooms, and it is notable that vented systems in particular are unable to excite the pressure mode properly in many rooms.

One of the reasons for this could be that there is a vent that allows the pressure to equalise. In theory, this is meant to be a resonant system, where the back wave of the loudspeaker is inverted in phase and augments the main cone wavefront. That the principle works is demonstrated by many large systems in auditoria, theatres and even outdoor venues, but all of these are large spaces where soundwave propagation is dominant.

When room dimensions become small compared to wavelength, soundwave propagation will not work, and bass can only be reproduced by pressurising (and de-pressurising) the listening space ... pressure mode. In tests I have performed in my workshop, a vented subwoofer seemed to make a lot of noise, but completely failed to produce bass that could actually be felt. A similar driver in a sealed box causes the whole house to vibrate, something that I have not been able to achieve to the same (or even similar) levels using any vented subwoofer system.

To some extent, this section is comprised of musings on my part, and at this stage I have no measurements to back up my claims. The effects I have reported are (or at least seem to be) real, but there is no clear point of delineation - I don't know exactly where soundwave propagation ceases to be the dominant force, and where pressure mode takes over. This will be covered in more detail when I get the chance to do some measurements and research.


6 - Subwoofer Positioning

One of the easiest ways to position a sub is to place it in the listening position - in the chair. Play material that has significant low frequency material, and then crawl around the room, placing your head in the most desirable potential locations. Listen carefully to the bass - it should be smooth and extended, with a minimum of large peaks or dips. The optimum position for the sub is now the location where you heard the best response.

It is very likely that the position of best response is completely undesirable for other reasons, so be prepared to spend a fair bit of time moving around, and listening carefully. There are always compromises, but with care you can still find a location that is acceptable aesthetically, is not inconvenient (e.g. the middle of the lounge room doorway), and does not cause howls of protest from SWMBO (She Who Must Be Obeyed).

It is a given that the other members of your family will naturally assume that you have finally lost it completely during this exercise, but it may be possible to get their assistance - or at least a second opinion. Involvement in the process could make it a lot easier to explain why the china cabinet really should be moved - preferably to another room if you have a powerful subwoofer


7 - Conclusion

It is probable that this article has not helped a great deal, and may even add to the confusion. Apart from the more common sealed and vented subwoofers, there are also dipole subs (of various types and designs), horns (which are generally too small, but often work well despite this), and even whole walls of low frequency drivers. No one system type can be recommended for all applications, and most drivers will function at their best in only one box type. Sub drivers that are designed for small sealed boxes will usually not work well in a larger vented box, and vice versa.

Ultimately, there are many possibilities, and it is up to the individual to make a decision based on their specific needs. Expecting a 50W amplifier with a 200mm driver to shake a typical lounge room is unrealistic in the extreme. Most of these units aren't really subwoofers at all, despite the maker's claims. Few households will have the space (or the need) for a pair of high power 450mm drivers in large enclosures driven with 300W or more for each speaker.

In between these extremes, there are many possibilities, and I can only recommend that you experiment and test likely candidates before outlaying a considerable amount of cash (to purchase a sub) or cash, time and effort (to build your own). Naturally, I suggest that you build your own if possible. This enables you to test each section, and make adjustments as needed.


 

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Copyright Notice. This article, including but not limited to all text and diagrams, is the intellectual property of Rod Elliott, and is Copyright © 2004. Reproduction or re-publication by any means whatsoever, whether electronic, mechanical or electro- mechanical, is strictly prohibited under International Copyright laws. The author (Rod Elliott) grants the reader the right to use this information for personal use only, and further allows that one (1) copy may be made for reference. Commercial use is prohibited without express written authorisation from Rod Elliott.
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