The effect of volume change on the low frequency response of a bass reflex or vented cabinet has popped up so I decided to do a rudimentary analysis on that using BassBox Pro, version 6.   Below are 4 alignments for a 10 inch woofer.  The internal volume decreases by 5%, 10% and 20%.  Below the 4 alignment charts are the plotted curves for several parameters.   All 4 alignme4nts are plotted on each graph, despite that on some, it may be hard to see each trace.

In short, the internal volume of a bass reflex /vented enclosure can be diminished by as much as 20% without much change in aural performance.  This will be elaborated upon in more detail later.

Keep in mind that this speaker design program, as good as it may be,  has NO ears.  Also, Thiele-Small parameters are derived using what is called small signal analysis, i.e. about 1 volt or less applied to the speaker which is hardly representative of that which is applied while listening to anything but very quiet music.  To be more realistic, Klippel analysis is more representative of normal listening as much higher voltages are applied to the speaker.  However, the data obtained from such analyses is useful mainly to manufacturers and high power users such as rock groups and/or outdoor sound reproduction who can drive speakers to their limits.  I might add home theatre aficionados also as they can drive speakers very hard. 

I might add that while many with whom I talked over many decades will consider the effect of bracing and the vent on box volume, NONE have mentioned the volume of the driver itself.  Consider this example of the optimum box of 3.57 cubic feet using a 12 inch woofer with a large magnet assembly housing a 2.25 inch diameter voice coil.  The frustrum of the cone occupies 103 cubic inches.  Add to that the volume occupied by the magnet assembly, about 44 cubic inches.  This unit has a die cast frame which occupies about another 7 cubic inches.  The total, 154 cubic inches.  This is 1.7 times the volume occupied by the vent assuming a 0.125 inch thick vent tube and 1.12 times the volume of 3 internal braces of 1.5 inch square stock.  

 

 

 

 

 

 

All cabinets use a nominal quantity of damping material, about an inch thick on all inner panels.

All vents are minimum size required to allow maximum diaphragm excursion.

Looking at the box parameters at the bottom of these charts, the following relationships can be seen.

Fb, the frequency at which the box is tuned, increases by about 0.5hz for each 5% decrease in volume.

F3, the -3dB knee is increasing at at a similar rate.  

Notice that the deviation in both these parameters in NOT linear.  This can be seen by comparing 1 to 2 and 2 to 3.  The box volume decreases linearly by 5% but the change in Fb and F3 seem to increase at an increasing rate.  There aren't enough samples taken here to verify or quantify that and such determination is moot as the box volume will become too small to justify the sensibility of such a determination.

There is no change in QL, which is leakage loss.  (vent)

 

 

Of these 4 vented box alignments, this is considered to be the optimum for this speaker. This is 95% of the optimum. This is 90% of the optimum. This is 80% of the optimum.
1  white 2  red 3  yellow 4  fuschia

 

 

 

 

 

Changes in some of the various parameters of the system as a result of box volume changes.  

Some are hardly affected. There are 4 curves in each graph.

 

Normalized to 0dB for box response comparison

 

SPL at 1m with specified power input,

in this case, 350w

 

Maximum SPL within maximum excursion, Xmax or thermal limit, whichever comes first, usually Xmax

It can be seen here and in the next set of curves that the most linear output of the speaker would be with an input power just a little under 100 watts.

 

Power required to reach Xmax or thermal limit

 

Cone displacement for specified input power, 

in this case, 350w relative to frequency.

The darker parts of the curves show excursion beyond Xmax

With 100w applied, the curve would drop about 2mm but maintain a similar shape, and won't exceed Xmax until 25hz

In this case, 350w, Xmax is exceeded below 31hz and between about 42hz and 70hz.

Compression at those points is inevitable and mechanical and/or thermal damage is imminent.

Fortunately, as can be seen from the impedance curves, the impedance in these areas rises, thus reducing the power transfer.

 

Simply the velocity of the air in the vent.

If the velocity exceeds 10% of the velocity of sound, spurious sounds may be heard, sometimes called whistling.

 

System impedance.

The left is that of the vent and the right, the speaker.

Fb is at the lowest point between them.

This assumes nothing between the amplifier and speaker,

crossover, L-pad, Zobel and the like.

 

 

Phase difference, expressed in angular degrees between the input signal to the speaker and the pressure wave emanating from the diaphragm.

A straight line would be ideal but wishful thinking.

 

Delay, expressed in milliseconds between the input signal and pressure wave.

 

 

 

 

 

 

This graph shows the changes in F3 and Fb, the former being the knee4, -3dB point in the rolloff; the latter is the frequency at which the box is tuned.

The left (Y) axis id frequency and the bottom (X) is volume.  The volumes here are decremented by 5%; the one on the extreme right is 20% smaller that the one on the extreme left.

It shows that both Fb and F3 are increased by only 2hz and 3hz, respectively.

The alignment for the 4th, 3.04 ft^3 was added to plot a smoother curve.

 

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