I have observed considerable confusion about the power requirements for a subwoofer. This is not surprising since the interaction between mechanical and electrical parameters of the subwoofer is complex and makes general statements very difficult. I will use the example of the THOR in conjunction with the spreadsheet closed-box1.xls to illustrate how the power amplifier and the maximum excursion capability of the driver limit the sound output. A similar analysis has also been done for a dipole woofer.

The analysis here is for the Peerless XLS 830500 driver, but any other driver data can be used in the spreadsheet. In all calculations I use Xmax = 12.5 mm. Let's start with the driver mounted in a sealed box of 47 liter (1.7 ft³) internal volume and driven from an amplifier with 75 W output capability into 8 ohm. For a sinewave signal this corresponds to a 34.5 V peak voltage and the amplifier, therefore, has probably +/-36 V supply rails. Depending upon the transformer and storage capacitor sizes in the power supply the output voltage will not vary significantly when the load impedance is changed. Ideally this same amplifier would have capability of 150 W into 4 ohm, and 300 W into 2 ohm. This would mean that the output devices can deliver 4.3 A peak into 8 ohm, 8.6 A into 4 ohm, and 17.2 A into 2 ohm. In practice the current capability of the output stage is limited and the 75 W amplifier might have a spec of 120 W into 4 ohm, not 150 W.

Figure 1 below shows the maximum SPL that the driver could generate at 1 m in free-space for 12.5 mm peak excursion, and the SPL it will generate when driven from the 75 W amplifier at different frequencies. The actual power delivered has the dimension of Volt-Ampere not Watt, since voltage and current are not necessarily in phase. Starting at low frequencies note that less than 75 VA is needed to obtain Xmax.

The power drops even further as the 36 Hz resonance frequency for the driver in the box is approached. Due to the increasing terminal impedance near resonance the fixed 34.5 V of the amplifier does not drive sufficient current through the voice coil and the SPL starts to deviate from the maximum possible. The sound output has become limited by the amplifier voltage. Above 59 Hz more than 75 VA are consumed as the current increases with decreasing terminal impedance. Around 80 Hz the 120 W into 4 ohm spec of the amplifier is reached. At higher frequencies the power demand goes as high as 165 W when the SPL follows the constant amplifier output voltage. The amplifier may not be capable of this current and the SPL could become current limited in addition to being voltage limited already.

The obvious solution for increasing SPL is to use a larger power amplifier, Figure 2. Going to an amplifier with 150 W into 8 ohm increases the output voltage by 3 dB to 48.8 V peak. Correspondingly the available SPL increases up to 3 dB above 22 Hz. The general shape of the curves is the same as for 75 W with current limiting potentially controlling the maximum SPL above 72 Hz for an amplifier that is specified at 225 W into 4 ohm.

It is interesting to observe the effects of a smaller enclosure, Figure 3, and a larger one, Figure 4. Both cases are for a 150 W amplifier. The resonance frequency increases to around 46 Hz due to increased stiffness of the 25 liter box volume. This requires more amplifier power below resonance, but less above it. The maximum obtainable sound output is essentially identical to the 47 liter box in Figure 2.

The same driver in a 470 liter box in Figure 4 has its resonance around 20 Hz and requires little power at low frequencies to reach maximum excursion of 12.5 mm, but above resonance it takes more power than for the 25 liter or 47 liter boxes in Figure 3 and Figure 2. The reasons for the differences in power requirements of the three different box sizes are not immediately obvious. One cannot make a flat statement saying that a smaller box requires more power. It depends upon which part of the frequency range one is talking about. The same goes for the larger box. What can be stated in general are the following limitations to maximum sound output:

The woofer's Xmax limits the maximum SPL at the low end of its frequency range. Amplifier power is rarely an issue.
Going up in frequency the amplifier is likely to have a clipped output voltage. The driver impedance is still relatively high in the frequency region above its mechanical resonance. The voltage, required to drive the current that is necessary to reach Xmax, is not available from the amplifier's power supply.
At the high end of the woofer's frequency range Xmax can typically not be obtained, because the current that can be delivered is limited by the safe current capability of the amplifier's output devices and the power supply's storage capacitors. The driver impedance has a minimum in this frequency region and even when the amplifier's output voltage is within its range, the current becomes limited.

The ideal amplifier would have variable supply rail voltages, so that it can deliver large current when the voltage is low and only needs to supply high voltage when the current demand is low.

The Peerless XLS drivers have a very strong motor and it is interesting to see what would happen if the flux density was less, so that Bl = 10 N/A instead of the specified Bl = 17.6 N/A for the 830500. With all other parameters unchanged the Qt now increases to 1.1 in the 47 liter box. Practically at all frequencies more power is required to reach the same SPL as in Figure 2.

It should be clear from the examples that amplifier power is not an issue at the low frequency end of the subwoofer range, but at high frequencies. The lowest frequency output is driver excursion limited. The high power requirement above resonance is due to the driver's moving mass that must be accelerated.

Equalizing the frequency response of a woofer with a "Linkwitz Transform" does not require any more power over what the driver can handle for maximum excursion without damage. Such equalization merely makes it easier to run the driver voice coil into its mechanical stops. Whether such equalization is useful depends solely upon the driver's volume displacement capability and the desired sound levels.

Electrical and mechanical limits to the maximum sound output from a closed box woofer

Electrical and mechanical limits
to the maximum sound output from a closed box woofer