There is a little box that nobody can believe in.
It's supposed to be a real shelf box, and therefore no deeper than 22 cm and no higher than 30 cm. At least that's how plus/minus European shelves are built. This means that a size of 30x30*22 cm^3 would still be pretty reasonable, resulting in a maximum internal volume of around 15 liters.
Furthermore, the frequency response should reach well down, 40Hz is the minimum. After a lot of back and forth, it is not possible to set up a bass reflex in this small volume. The port is always either too long or too narrow - resonances or wind noise. Passive membranes don't help either, because where to put them on the shelf, and they would also be much too heavy and, last but not least, expensive.
What remains is a sealed design with the largest possible cone and plenty of excursion. A two-way speaker won't work, because the excursion would lead to distortion of the midrange in any case, no matter how fancy the bass is.
So three ways with the lowest possible crossover frequency. This can practically only be done (digitally) electronically. Furthermore, the necessary bass correction can again only be achieved electronically.
This then leads almost inevitably to the design as shown. Three ways, bass on a separate amplifier with boost, bass / mid digitally separated, mid / high passively separated on a second amplifier. A quick calculation shows that 50 watts each will exploit the limits of the speaker fully. The digital separation and smoothing of the frequency response can be done via the PC / Equalizer APO / Peace and a 7.1 USB interface; 2 channels stereo to four channels speakers. With 24-bit digitization depth, volume control on the digital side is completely sufficient. (This ensures synchronization).
Since this is an experiment, fairly inexpensive drivers are sufficient. The bass allows +/-6mm travel. And the result is actually really good.
The key data: Cabinet made of 9mm poplar plywood, 21 x 30 x 21 cm^3, stiffened inside, approx. 8 liters, midrange 0.5 liters, drivers from Peerless and ??, x/overs at 300Hz and 2kHz. The bass is digitally tuned to a cut-off frequency of about 30Hz in the room.
I have a question: how can I prove that the choice of a three-way system is advantageous? I am not aware of any examples of a similar design.
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