I’ll introduce two studies I performed for this kind of systems;
1. Alpair 5 + 13cm Woofer Dual Amplifier System
Fig.1: Alpair 5 + 13cm Woofer
Data:
Main speaker: Mark Audio Alpair 5 metal cone full-ranger, sealed enclosure with 1L volume
Woofer: Dyanavox LW5002PPR-S PP cone woofer, sealed enclosure with 3L volume
Fig.2: System Schematics
Fig.3: Analog Channel Divider: Behringer CX2310 Super-X Pro
Fig.4: Responses of the woofer with different Cut-offs
The basic response of the woofer starts rolling-off at approx. 100Hz with approx.12dB/Oct slope. This is a typical sealed box behavior with plenty of absorption materials inside (which effectively suppresses mechanical resonances of the driver-box system). Additional three curves in the figure are measured with different cut-off settings; 44Hz(the lowest position), 60Hz, 120Hz with -24dB/Oct High-Cut filter. Since the basic response has +12dB/Oct slope in the frequency range of <100Hz, the filtered response curves decay with -12dB/Oct slope within this frequency range (because +12 -24 = -12dB/Oct). This is quite convenient to combine with a sealed boxed main speaker.
Fig.5: Combination with Alpair 5
Now, a curve from Alpair5 (1L sealed also with absorption materials inside) is added in the figure. This also rolls-off with 12dB/Oct slope. To get a flat response in the cross-over range, both curves(i.e. woofer and Alpair 5) should cross at -6dB point. The frequency of this point is fixed by the main speaker (i.e. Alpair 5 in this case, and the cross-frequency = approx. 100Hz). To achieve this, the Cut-Off freq. and the gain balance between two speakers should be selected carefully. Finally, the Cut-Off of 60Hz is selected as the optimum shown as Fig. 5. For this kind of configurations, it is important to understand that the filter’s Cut-Off freq. and the actual Cross-Over freq. are not equal because the basic response of the woofer is not flat.
Fig.6: Total frequency response
The finally obtained full range frequency response is shown in Fig.6. Quite enough low-end performance (30Hz/-10dB or more) can be achieved with well flat cross-over range. The low-end limit (roll-off freq.) of this kind of system does not depend on the woofer size, because any sizes of woofers would have the same 12dB/Oct response in <100Hz range if they were installed in a very small sealed box fully stuffed with absorption materials. If you use a smaller woofer, you will need to increase the amplifier output accordingly to balance with the main speaker. To prove this, I just tried Alpair 6M (3” full-ranger) as the woofer.
Fig.7: Alpair 6M used as the woofer
Green lines indicates the result with Alpair 6M. As you can see, identical low-end response can be achieved even with a much smaller driver. With traditional passive network configurations, you must use larger woofer to bring the low-end limit lower. So, Low-end limit basically depends on the woofer size. With those systems, you need a very big woofer if you want to have 30Hz/-6dB response (usually 38cm woofer + >100L box will be required, as shown in Table.1).
Table.1: JBL Monitor Series
In case of powered woofer systems, the woofer size just depends on the sound volume you need. if you live in a small room (please understand Japanese standard environment), or if you don’t need tremendous sound volume, or if you want to have a near-field / desktop system, and if you can accept an additional power amplifier and a divider, you can reduce the system size drastically with achieving equivalent or even better low-end response comparing with full or mid sized conventional bas-reflex systems. Further, needless to say, bass generated from a sealed box has much better quality in nature.
Fig.8: Possible Configurations
Fig.8 is showing several possible configurations with 3” full-rangers. For example you can use Alpair 5 or 6 for this purpose. As the 4” woofers, CHR-70 might be useful for instance. Alpair 10 woofer must be the optimum due to its big Xmax. All these configurations have an identical frequency response (30Hz/-10dB or more) as mentioned above. Again, the low-end limit does not depend on the woofer size at all. You should select the size just depending on the sound volume you need. For desktop/near-field applications, MINI or SMALL could be sufficient. Actually, my new desktop system with Alpair 6M named "ZAP" (see next article) belongs to SMALL class. For typical Japanese private rooms, COMPACT or MEDIUM may be enough, if he is not a big sound enthusiast. My "Alpair 5+13cm woofer" system belongs to COMPACT class. This is rather bulky for desktop applications.
More than 80% of music listeners feel comfortable with <80 dBA sound pressure level at their ear position, according to a research performed by a Japanese authority. If you listen to music routinely with more than 85 dBA sound pressure, your ears will be damaged, some European agency says. If you listen to music within these moderate sound levels in your private room, and you feel full-sized systems as unnecessarily bulky, however, if you want to have real bass that is enough to enjoy syomphonies in deep (hopefully good response down to 40Hz), these systems must be quite attravtive. You can select your best size as needed without sacrificing bass performance. I believe this must be a very basic requirement for any music reproducing machines.
Currently, very cheap and compact digital amplifiers with good enough sound qualities are available (e.g. Nuforce Icon). So, cost of additional amplifier seems not critical. Cheap and compact 2ch digital channel divider with integrated DAC could be quite useful for this kind of applications. I want it !
2. Minimum 2.1ch System
I made a very nice minimum 2.1ch sound system named “KERO” based on the theory as described above. This can be classified as MICRO that is smaller than MINI in Fig. 8.
Fig.9: Micro Monitor KERO-1
Fig.10: Schematics of KERO
Data:
Main speaker: Aura Sound NSW1-205 1” metal cone full-ranger; sealed enclosure with <100cc volume
Sub-woofer: Aura Sound NSW3-193 8A 3” metal cone full-ranger; sealed enclosure with approx. 700cc volume
Sub-woofer Amp.: Victor 30W, integrated variable High-Cut filter and fixed 50Hz Low-Cut filter for safety
Response: 45Hz – 20kHz/-6dB
Fig.11: Response of KERO
This system is targeted for super near-field applications within 50cm distance. As you can see, very good low-end performance is achieved even with such minimum size (total volume < 1L !). This system sounds quite clear and natural with quite sufficient bass much more than I expected. I’m using KERO beside a folding bed in my office room, with iPod Classic and a small tube amplifier (6BM8 single). Placing KERO just behind the pillow, I can enjoy very clear sound just like as head-phones because of substantially no distrubing room effects. This is really super!
Fig.12: KERO on site
5) 30Hz
music sources(from left);
music sources(from left);
Base level of each setting is determined so as to 
Calculation results;
