New loudspeaker plans... Part B

In an earlier post I started writing up some plans for a loudspeaker that I want to build...

To summarize, it's a 3-way design with a somewhat classic choice of driver sizes: 25cm (10") woofer for the bass, 12cm (4.5") midrange, and a fairly standard 25mm (1") dome tweeter. They will have sealed enclosures so I won't have to deal with issues like temperamental tuning and a peaky bass response. For the woofer I looked at the idea of "pressure loading" (or at least, that's what I'm calling it), whereby the box is small enough so that air resonances only occur at frequencies outside of the speaker's operating range.

The midrange

Before continuing with the box design for the bass, I need to start thinking about what to do for the midrange speaker. Firstly, I have a feeling that a pressure-loaded box won't be a good idea. Given an estimated cut-off frequency of 3~4kHz, the half-wavelength will be around 3~5cm, which is tiny! The air volume behind the midrange speaker would have to be less than one litre, so let's do some quick estimates and go from there:

Half wavelength at 5kHz:

λ/2 = 340/5000/2 = 34mm

Given a cone area of 50cm² (I'm using the Seas L12RCY/P for this example) the effective radius is:

r = √(50/π) = 40mm

A cylindrical volume that is slightly wider than the cone would be approximately:

[ π*(34mm + 40mm)² ]*34mm = 584914mm³ = 584.9cm³ = 0.58L

According to Subwoofer Simulator, when I specify 0.58L as the box volume for a Seas L12RCY/P, the system forms a mechanical high-pass filter with a cut-off at approximately 150Hz. That is decidedly annoying because it's fairly close to the 200~300Hz cut-off that I want. It means that if I want to electrically filter out some of the bass, then the combined filter response will have to be at least 3rd-order and its accuracy will be highly dependent on mechanical and environmental factors such as the speed of sound on a particular day.

Another thing that I'm not sure about is: what happens to the cone when it's cushioned by a relatively small pocket of air? Maybe it will flex and resonate at substantially lower frequencies than predicted in tests that use large air volumes? Maybe the air suspension will be sufficiently affected at low frequencies that it cause inter-modulation distortion when combined with other frequencies?

I think that the midrange needs a relatively large box instead, together with a highly effective means of absorbing resonances.

Anechoic wedges

What better way to describe my idea than to draw it?

What I originally had in mind was a sort of "reverse horn" where the sound waves gradually become more and more concentrated as they radiate away from the speaker. Eventually the energy becomes so concentrated that the surrounding enclosure turns into a heat sink. That idea evolved from a cumbersome spiral-shaped horn into an array of miniature wedges – it's basically still the same thing but in a different format.

Part C: enclosure materials, crossovers, and cohesion between bass and midrange.

New loudspeaker in the works... Part A

Well, I'm designing a new pair of loudspeakers. Basically, they'll be a pair of traditional 3-way loudspeakers that use 25cm (10") woofers housed in sealed boxes, 12cm (4.5") midrange speakers, and 25mm dome tweeters. (The original web page that I started about them can be found here.)

I don't intend to build these loudspeakers any earlier than 2008 – partly because I'm a great procrastinator and partly because I don't have anywhere to put them at the moment. So in the meantime I can do the fun part: explore various options, possibilities, and do calculations for the design...



I chose a classic 3-way style because:

1. It can use relatively benign crossover frequencies around 200~300Hz and 4~5kHz.
2. The ratio of driver sizes is quite evenly distributed, allowing a predictable off-axis response.
3. From past experience, a 25cm (10 inch) woofer in a sealed box would have quite well-balanced performance considering the effects of room gain at low frequencies and its ability to reproduce frequencies up to 200~300Hz.
4. Unlike many other designs such as 2-way and quasi 3-way loudspeakers with large midrange drivers and low crossover points, none of the drivers are pushed to their limits. This means lower distortion and higher power handling.

So if all goes well, the design might even turn out quite elegant due to the lack of any glaring weak points.

Drivers

As a starting point, a suitable woofer may be a Seas L26RFX/P. Midrange units: Seas L12RCY/P, Excel W12CY001, Visaton AL 130 8ohm, or Eton 4-300/25 Hex. The tweeters may be chosen at a later stage.

Box design

Tweeters

Practically all dome tweeters are sold with sealed chambers already built in – either behind the magnet or as a small pocket of air directly behind the dome – so designing a box for them is almost a non-issue. There are some things to remember however: mechanical vibrations and diffraction of the wave-front around nearby box corners. Those effects may produce resonances or adversely affect the off-axis response.

Woofers

The enclosures will be constructed from high grade plywood. One option that I'm interested in is the use of small pressure-loaded boxes.

Pressure-loaded, high Q enclosures:

Although the unequalized bass response may be less than optimal, there could also be several advantages over a classic design where the Qtc is tuned to 0.707:

• The idea is that if the box is sufficiently small, there won't be any resonant air modes throughout its entire operating range.
• A small box will be less reliant on the linearity of the speaker's spider and surround – the relatively high spring constant of the air will dominate.
  
• Extra space and convenience.
  

One way to work out whether or not such a design is feasible is to decide what the minimum resonant frequency should be and to calculate corresponding dimensions from that. For example, let's say that the lowest allowable resonant frequency is 500Hz. From that we can work out the wavelength, from the wavelength we can work out the length of a resonant column of air and therefore the maximum dimensions of a box.

Given a sound velocity of approximately 340m/s (it varies), the wavelength is:

λ = 340 / 500 = 0.68m

But, what fraction of the wavelength is relevant? A quarter? Half? A whole wavelength? And why? This is where a bit of nodal analysis comes in handy. (A refresher: nodes are regions of minimum velocity and maximum pressure oscillation, while anti-nodes are regions of maximum displacement and minimum pressure variation). A loudspeaker can be likened to a musical instrument where the speaker driver is like a resonator or an actuator of some sort, such as a reed, while the box forms a resonant column of air, similar to a stopped organ pipe.


My initial line of thinking was that there'll be an audible resonance when the length of an air column is one quarter of a wavelength. However, I looked around on the web and people's measurements suggested that the system would behave as though both ends are fixed, not just one end. At first I wasn't quite sure why that was the case.

I tried to visualize in my mind what happens and then I realised that when the speaker cone is at an anti-node, it directly controls the displacement of the air. It's not really a resonance at all if the cone directly controls the region of maximum displacement.

When it's at a node, there is no such control. It means that there's at least one anti-node floating inside the box, which could reach an enormous displacement amplitude if there's very little damping. The node next to the speaker cone therefore produces a large opposing force and that is what I want to avoid.

For a 500Hz frequency limit, the maximum allowable length is therefore:

L = λ/2 = 0.34m

Rough estimate for a cube shaped box:

V = 3.4dm³ = 39.3L

A slightly more accurate estimate would be to calculate the distance from the centre of one box face (where the speaker is positioned) to a corner on the opposite side:

B = √(A²/2)
C = √(A²/2 + A²) = √(1.5*A²) = 0.34m
A = √(0.34²/1.5) = 0.278m

V = 2.78dm³ = 21.4L

The resulting volume is a bit on the small side, however a cube isn't exactly the most efficient possible shape. How about a hemisphere? Using the formula for the volume of a sphere:

V = 4/3πr³

where r = 0.34m or 3.4dm, the volume of a hemisphere is:

V_h = 2/3πr³ = 82.3L

Obviously the calculations are just estimates that ignore lots of real-life variables, but they're accurate enough to give some ballpark values that could be used in a process of elimination. A box somewhere between 20L and 80L, with no resonances up to 500Hz certainly sounds appealing. The trick will be to devise an effective shape that doesn't clash with other requirements such as minimizing the distance between the woofer and the midrange driver.

Check back soon for Part B where I'll look at some other options: over-sized boxes, anechoic wedges, "reverse horns" and similar possibilities for the midrange enclosures.

Sealed boxes

A basic sealed speaker box usually consists of a hollow box and a large circular hole where the speaker is mounted.

At low frequencies, the air and damping materials form an energy storage device – a mechanical spring – which produces an unavoidable resonance when combined with the spring constant and mass of the speaker cone. This resonance is usually exploited to optimize the low-frequency response of the loudspeaker.

At relatively high frequencies (e.g.: above 500Hz), the wavelength of the sound is short enough to produce various echoes and partial oscillations inside the box. Its purpose then is to absorb as much of that energy as possible. Damping materials such as pillow stuffing are often used, as well as design techniques such as an acoustic labyrinth.

A compromise has to be made between 'tuning' a box volume for good performance at low frequencies, or making it very large in order to reduce the amplitude of midrange resonances. Many designs make the enclosures relatively small because of space constraints. Furthermore, smaller boxes often have practical and structural advantages over relatively large boxes.

A sealed loudspeaker can be modelled as a second-order system at low frequencies. It has:

• One mass – the speaker cone (in parallel with part of air mass inside the box).
• One spring – the spring constant of the air, in parallel with the speaker suspension.
• One damping device – electromechanical damping of the speaker (Qms in parallel with Qes).
  

Box design

Designing an enclosure is a fundamental part of overall loudspeaker design. It's nowhere near as simple as building a "strong, solid box". A speaker box performs a variety of functions and some of them are actually not so obvious.

Take for example: a speaker that's framed in a sealed hollow box, which has nothing but air inside it. What happens? How does it work? Why is the box necessary? Is it necessary?

In order to decide what kind of loudspeaker system you want to build, you'll also need some background knowledge and understanding of how various designs work. Anyway, check out some of my other posts regarding sealed, vented/ported, and labyrinth designs.