Vented Boxes

Vented box design is an exercise in two degrees of freedom. You decide how big the box and what frequency it is tuned to and that determines the response shape. Below are some simple curve fit relations from design charts. These relations are still a great starting point for deciding how big of a box to make given the parameters Fs, Qts, and Vas.

From Small-Margolis 1981 Journal of the Audio Engineering Society.

Vb = 20 * Vas * Qts^3.3

Fb = 0.42 * Fs * Qts^-0.96

F3 = 0.28 * Fs * Qts^-1.4

From D.B. Keele Jr. (citation unknown, but found in several books by David Weems)

Vb = 15 * Vas * Qts^2.87

Fb = 0.42 * Fs * Qts^-0.9

F3 = 0.26 * Fs * Qts^-1.4

To tune a volume Vb (in liters) to a frequency Fb (from Small-Margolis again)

Lv = 2350*Dv^2/(Vb*Fb^2) - 0.73*Dv

Lv and Dv are vent length and diameter in millimeters, Lv is assumed to be larger than 50mm for this formula, so if it comes out shorter, pick a larger Dv.

If you measure Fb and find it to be different from the predicted value, you can figure out how much to add or subtract by calculating:

deltaLv = -deltaFb*Dv^2*2350/(Fb^2*Vb), where deltaFb = Fb_desired-Fb_measured.

Minimum vent diameter in mm

Dvmin=10*sqrt(10^(SPL/20)/(PI*Vmax*Fb))

Where:

Dvmin is the minimum vent diameter in mm

SPL is the maximum SPL from the vent

PI=3.14159...

Vmax is the maximum desired port air velocity (typically <17m/s)

Fb is the tuning frequency

The only gotcha with this formula is you need to pick a realistic SPL to expect from the port, which is best informed by a program that can calculate Max SPL. In a pinch, you could also just use SPL = dB/1W + 10*log(Max Power).

Here is a picture that illustrates how port diameter varies with Fb and desired SPL. Plotted are 90-120dB in 10dB steps. If you had a sub capable of 120dB (green curve) at 30Hz, you would want a 250mm (10") diameter port. Also note that a 100mm (4") port will do 100dB at 20Hz.

Changing SPL requirements by 6dB up or down will double or halve area, and changing by 12dB will double or halve diameter. From the example above, If you wanted to do 112dB at 20Hz, you would need a 200mm (8") diameter vent, or 4x 100mm ports.

Passive radiators

For a passive radiator, use the piston diameter as the vent diameter in the formula below and find

Lv = 2350*Dv^2/(Vb*Fb^2).

The total mass of air in the vent (in kg) is then

M = 9.27e-10*Dv^2*Lv

The mass to be added to the passive radiator is

Ma = M - Mmp, where Mmp is the rated moving mass of the passive radiator, again in kg.

Passive radiators have a notch in frequency at the PR's "free-air" resonance frequency. This is altered by the added mass and must be calculated.

Fnotch = Fp*sqrt(Mmp/Ma), where Fp is the original PR resonance and Fnotch is the new resonance.