Rohit Balkishan Dubla
A woofer or sub-woofer typically needs a proper enclosure for it to perform optimally. The enclosure requirements for a mid-range driver are usually not so critical since they are supposed to only work with a band-limited signal and usually have a flat-enough response within this range, but the enclosure design process can be applied to a mid-range too. Designing a loudspeaker enclosure is as much an art as it is a science. There are a few trade-offs to be made and the final result may require a bit of tweaking to get it right.
Driver parameters
The first step to designing a suitable enclosure for a given speaker driver is determining the driver's Thiele/Small (T/S) parameters. Some manufacturers may provide this and for others we have to measure it. Measurement is recommended even if parameters are published given that between units of the same driver there will be quite a bit of deviation from the published figures.
Refer to the ESP article on T/S parameter measurement for detailed steps to do the measurement.
Once you have followed the steps in the above page, you should have the following T/S parameters for the driver:
Free-space resonance frequency, Fs (Hz).
Electrical damping, Qes (unit-less).
Mechanical damping, Qms (unit-less).
Total free-space damping, Qts (unit-less). Qts = (Qes * Qms) / (Qes + Qms)
The driver's Vas (in litres or cubic feet depending on prevalent measurement standard).
Note: Q indicates how (under) damped a system is, and damping is inversely proportional to Q. Therefore as Q increases, damping decreases (i.e., the system becomes more under-damped). Qts is also known as the driver's "resonance magnification" - this implies that as this increases, the driver's damping (at resonance) decreases.
In addition to the above, for vented design the following parameters must be known too:
The driver's effective cone diameter (D) or radius (r) (in mm or cm or inches depending on prevalent measurement standard). This is usually the cone's diameter plus the width of the surround on one side. A rough calculation is simply 80-85% of the driver's frame diameter.
The driver's peak linear excursion, Xmax (in mm or cm or inches depending on prevalent measurement standard). This is tricky to determine - a wedge micrometer might be used to determine this. However we cannot be sure when the exact maximum linear travel is reached, so some guesswork may be needed. An easy way might be to simply push (or pull) the cone till it cannot move easily anymore (be careful not to physically damage the driver!) and measure that distance. Estimate the Xmax to be 75-80% of this value. As a rough guide,
Cone diameter | Approx. Xmax
--------------|--------------
6.5" (16.5 cm)| 4 mm
8" (20 cm) | 8 mm
10" (25 cm) | 10 mm
12" (30.5 cm)| 12 mm
Note that the above table is only a rough guide and might not apply to the particular driver you may be using.
With these in hand, you can determine the optimum enclosure for the driver. Linearteam's WinISD is a great freeware that makes enclosure design a breeze. The manual process is described below and is very helpful as a learning exercise. Note that the formulas in this page assume metric units (mm or cm for length and cc or litres for volume) for measurements. Substituting imperial/non-metric measures without first converting them to metric will cause incorrect results.
Before proceeding, the efficiency-bandwidth product or EBP of the driver indicates what type of enclosure the driver is suited to. The EBP is calculated as,
EBP = Fs/Qes The following shows the suitability of the driver for a given enclosure:
EBP < 50: Sealed
EBP 50 – 90: Sealed/Vented
EBP > 90: Vented
Note that the EBP is only an indicator and it does not mean that EBP absolutely dictates the enclosure to use. As will be clear from the following paragraphs, drivers may fit one, or both or none of the enclosure types.
This is by far the simplest to design. Keeping in mind that when put inside a box, the resonance frequency and Q will both be always greater than the driver's corresponding free-space T/S parameters, we can proceed in two ways, viz., derive a box volume for a certain target Q (Qtc) or choose a box volume and determine what Q and resonance frequency we will get. The former is usually a better starting point since Qts is already known which makes choosing a suitable Qtc easier.
Depending on the Qts, Vas and Fs for the particular driver we may be able to design a box that has a flat or peaky response. This is the compromise or trade-off involved in the design of any loudspeaker. Drivers that have a low Qts, Vas and Fs allow the designer to target any kind of response, but the physical construction of the driver makes these parameters inter-dependent - changing one will result in one or the other (or both) parameters to change, and also influences the cost of the driver. For example, a heavy cone will result in lower Fs but increased Qts. To then lower the Q as well, we need to either make the suspension stiff (Fs increases) or increase the magnet strength (probably greater manufacturing cost).
A Qts > 0.6 implies a large box (if good low-frequency response is expected). It turns out that such drivers also have a relatively large Vas (as compared to a driver with low to very low Qts), and the box volume is directly proportional to Vas for any Qtc, as the following equations show.
In any case, setting Qtc to much more than 1.5 (and therefore a very small box) will result in a speaker that has a very large resonance peak and may render the speaker to sound as if it has a narrow bandwidth, and a sort of one-note sound. Similarly, a very low Qtc (< 0.5) will result in a box that is over-damped and the loudspeaker will sound weak or lacking in bass. Where the Qts allows (i.e., < 0.7) we should aim for a target Qtc of 0.707 (maximally flat response before cut-off). However, depending on the Fs and Vas, the resulting box volume and/or lower -3 dB cut-off might not be as desired (box volume too great or cut-off freq. too high). In this case we choose increasing Qtc till the box volume and cut-off frequency are acceptable. A Qtc of 1-1.2 results in a box that has both small size and a slight peak that accentuates the bass. However this does not necessarily mean a low cut-off frequency, but the peak in response makes it sound as if it has deeper bass.
Designing for a target Q: Here, we designate the target or system Q as Qtc. Choose a value greater than Qts (else we will end up with negative box volumes). This is basically the driver's Q when put inside a box. Use Qtc and Qts to find α as,
α = (Qtc / Qts)2 - 1
The box volume is then given by,
Vb = Vas / α
Designing for a target box volume: Here, we choose a target volume Vb and calculate resulting total or system Q (Qtc). Use Vb and Vas to find α as,
α = Vas / Vb
The Qtc is then given by,
Qtc = Qts * √(α + 1)
For both the above cases, the following equations can be now used to find what the system resonance and cut-off frequencies will be for the particular box volume:
System resonance Fb = Fs * (Qtc / Qts)
The -3 dB cut-off frequency F3 can be calculated as,
k = (1 / Qtc2) - 2
k' = √(k2 + 4)
F3 = Fb * √[(k + k') / 2]
The peak gain (at box resonance where Qtc > 1/√2) is given by,
Amax = Qtc2 / √(Qtc2 - 0.25)
For Qtc <= 1/√2, Amax is simply unity or 0 dB.
As is evident from the sealed enclosure design process, not all drivers may lend themselves to work in a sealed box. These may require a box that is too large to be practical to achieve acceptable low-frequency range or might have an Fs that is simply too high to be used in a sealed box. Such drivers might benefit from a vented enclosure which allows us to use the vent's own resonance to extend the driver's low frequency response in the box. Needless to say, some drivers can be used only in sealed, some only in vented and some in both. There may also be drivers that seem to be unfit for either box type - such drivers have their damping (Qts) solely dependent on its suspension (typically > 0.7) and usually have a rather high Fs and must be fitted into an enclosure that is essentially an infinite baffle (or a very large box) - typically found in car speakers where they are fitted into the vehicle's boot. Their low-frequency response is extended due to the cabin-gain effect and sound as if they have a good amount of bass.
The vented enclosure design process is quite a bit complex and iterative in that we start with one or two target parameters (the -3 dB frequency, for example) and calculate the resulting box volume and port dimensions. In case the box and/or port dimensions are not acceptable, we repeat the same with a different -3dB frequency. Alternatively, we can change the box and/or port dimensions to see what -3 dB frequency can be attained. Note that only the 4th order high-pass ported design is covered below. This is the most common vented enclosure that we encounter and has one or two vents drilled into the enclosure's panel. The vent(s) may also have a tube whose length can be used to tune the box. A vent without a tube basically has an implicit tube whose length is the same as the panel's thickness.
Design for a target -3 dB cut-off: Assuming we start with a target -3 dB frequency (F3), calculate the box volume as,
Vb = Vas / [(F3 / Fs)2.273]
Design for an ideal box: Here, the box volume is calculated so as to give the flattest response with the lowest -3 dB frequency (F3).
Vb = 20 * Vas * Qts3.3 and the cut-off frequency can be determined as, F3 = Fs * (Vas / Vb)0.44
Once the box volume is known we can find the tuning frequency Fb as,
Fb = Fs * (Vas/Vb)0.31
With Fb known, calculate the minimum port or vent diameter as,
Dmin (in cm) = 2.030 * [(Vd2 / Fb)0.25 / √Np], where Np is the number of vents. Vd is the maximum volume displaced by the cone's excursion. This can be calculated as,
Vd (in cc or litres) = π * r2 * Xmax, where r is the cone's effective radius (in cm) and Xmax is the driver's peak linear excursion in cm (if specified in mm, divide Xmax by 10). This can be alternatively stated as
Vd = (π * D2 * Xmax) / 4, where D is the cone's effective diameter (in cm).
Note that Dmin is the smallest diameter that the vent can have. Any vent smaller than this will not have any significant tuning effect, and carries the risk of unwanted port noise. And finally, the port's length can now be calculated as,
Lv = [(2.35625e+4 * Dv2 * Np) / (Vb * Fb2)] - (k * Dv), where Dv is the port diameter (≥ Dmin), and k is the "end correction". A value of 0.75 is typically used for this and accounts for ports that have a flared opening. However for ports that are not flared, set this to 0 and while attaching the tube to the panel, factor in the panel's thickness such that the tube's length plus the panel thickness equals Lv. Lv has the same unit of measurement as Dv. Note that for multiple ports Lv signifies the length of each port.
As can be seen above, the ported design process is a lot more involved and multiple iterations may be needed to arrive at a suitable compromise between low-frequency response, box volume and port dimensions. It is best to use a loudspeaker design software for a vented box to minimise errors. This also allows one to quickly change box or vent parameters to see the kind of response that can be achieved. Also to be noted is that a vented enclosure is a lot more dependent on the driver's T/S parameters and unless these are determined to a high degree of accuracy the final response will not be as desired. This also implies that a vented enclosure is very specific to the driver it has - changing the driver (or even a reconed driver) might render the enclosure useless as far as tuning and frequency response is concerned. In this aspect, a sealed box is a lot more tolerant and much less severely degrading in case of a driver mismatch.