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Subwoofer design estimates

The most important factors to consider in the design of a subwoofer are volume displacement capability of the driver, acceptable cabinet size, and power amplifier requirements. The following provides some initial estimates.

1 - Volume displacement
To reproduce low frequencies requires the displacement of large volumes of air to obtain appreciable sound levels. The driver chosen for this project is the Peerless XLS 830500, a 12" unit with a cone area Sd = 466 cm² and a displacement of 12.5 mm peak, for a volume of 583 cm³. When placed in a sealed box, so that the speaker acts as an omni-directional, monopole source, it will theoretically produce a 20 Hz free-space sound pressure level of 90 dB SPL at 1 m distance, 102 dB at 40 Hz and 114 dB at 80 Hz. This can be calculated using the spreadsheet spl_max1.xls .
The SPL will increase when the woofer is placed into the confines of a room. Assuming 7 additional image sources due to a stiff and reflective floor, rear wall and side wall, the pressure will increase eightfold, or 18 dB, if all 4 sources are closer than 1/8th wavelength to each other, as when near a room corner. So the 90 dB SPL at 20 Hz of free-space is more like 108 dB in the room. There may be further increase due to room gain, if 20 Hz is below the lowest room mode frequency. If 20 Hz is in the sparsely populated mode range, then there may be more increase than desired at certain frequencies. Parametric equalization can alleviate this problem.
It is difficult to be precise, because the actual room behavior is almost impossible to predict, but I would conclude that using a single driver with this much volume displacement should provide worthwhile low frequency output. If not, then the addition of a second woofer will give a 6 dB increase, which is significant. It will also allow excitation of the room from two different locations, which can make for a smoother response, even when both woofers are driven with a mono signal.

2 - Box size and amplifier power
With the driver mounted in a sealed box the enclosed air acts as a spring and reduces the compliance seen by the motor of the driver. This raises the mechanical resonance frequency. With a specification of Cms = 0.46 mm/N, corresponding to the compliance of an air volume VAS = 139.2 ltr, and a free air resonance of fs = 18.1 Hz, when mounted into a box with an assumed internal volume Vb = 50 ltr or 1.8 ft³, gives an effective compliance: 
VAS' = VAS Vb / (VAS + Vb) = 36.8 ltr 
Cms' = (VAS' / VAS) Cms = 0.12 mm/N .
The resonance frequency is raised to:
Fb = fs sqrt(VAS / VAS') = 35.2 Hz
The resonance frequency falls right into the operating range that is intended for the subwoofer. That poses no problem, if the power amplifier can drive a wide range of load impedances. It rules out tube amplifiers, but solid-state amplifiers can handle this situation. 

More serious is the increase in stiffness, which dominates the driver behavior below the resonance frequency, and against which the cone has to move. A deflection of 12.5 mm requires a force F = Xmax / Cms' = 104.2 Newton or 23.4 pound. The motor has a force factor Bl = 17.6 N/A. Thus it takes Ip = F / (Bl) = 5.9 Ampere current to develop such force. The current flows through the voice coil resistance Re = 3.5 ohm. If sinusoidal, it causes a power dissipation of P = 3.5 x 5.9² / 2 = 60.9 W.
Amplifiers are usually specified for 8 ohm loads, so the 5.9 A peak current would require an amplifier with 139.2 W capability, which is still reasonable.

Above resonance the moving mass, Mms = 166.3 g, dominates the driver motion according to force = mass x acceleration. At 80 Hz the peak cone acceleration is a = w² Xmax = 3158 m/s² and F = 525 N. Thus the peak current becomes Ip = 525 / 17.6 = 29.8 A. It would take an amplifier capable of delivering 3562 W into 8 ohm to drive the 830500 to full excursion! Clearly it is not feasible to obtain 114 dB SPL at 80 Hz, even 104 dB still takes 356 W. The amplifier requirements for the THOR are not as severe, because its operating range is below 40 Hz. 

The above estimates assumed a box volume of 50 liter. Increasing this figure reduces amplifier power, but it becomes a matter of fitting into the listening room space, as to how far you can increase box size, especially when two units are desirable. 
I did not consider alternate approaches to subwoofer design as acceptable for meeting my goal of accurate sub-bass reproduction. This includes vented, passive radiator and acoustic bandpass woofers. They all rely on resonant energy storage to increase efficiency and to reduce size. 

The spreadsheet closed-box1.xls makes it easy to evaluate the effect of box size and driver parameters upon amplifier power and obtainable SPL. It will be used to refine the estimates and to correlate them with measured performance and frequency response equalization.

| Introduction | Estimates | Design | Measurement | Equalization | THOR-ORION xo |
| Supplies | SPL limits |