Thermal Compression
Subwoofers achieve deep bass response even though their enclosures are usually quite small in proportion to the driver cone area because they are equalized. Without equalization the response might extend only to 50 Hz or 40 Hz. The lowest frequencies are boosted to provide balanced system response to low frequencies.

A consequence of this boosting is that much more power is delivered to the driver than in a normal, unequalized speaker. The power supplied to the drivers at 20 Hz can be 20 or more times normal. This large amount of power results in the coils running hot even during only moderately high demands.

A significant problem stems from the fact that the electrical resistance of copper is strongly temperature dependent, having a temperature coefficient of 0.39%/°C. Therefore, when the voice coil temperature reaches 200°C, the resistance has increased by 70% which causes a reduction in current, and, therefore, output, by 4.5 dB.

The situation is actually more complicated because resistance is only one of three components of driver impedance (although the largest at most subwoofer frequencies), the others being motional and inductive impedance. The following graph shows the total impedance of the drivers in the enclosure and also the resistive, motional and inductive components that combine to produce it.

Total impedance of the SS2 (top line) and its components of resistance (at 0 dB), motional impedance and inductance.

Since the resistance is the only one of these three impedance components that is temperature dependent, the impedance does not increase with temperature uniformly at all frequencies. Where the other components are major contributors to the total impedance, near resonance and at higher frequencies, the impedance does not increase as much as it does at other frequencies.

Therefore, the shape of the total impedance curve changes and therefore the frequency response changes. The following graph illustrates, from top to bottom, the impedance at 200°C, the room temperature impedance, and the change in frequency response that results. You can see that the sensitivity has decreased 2.5 dB at 40 Hz but 4 dB at 20 and 80 Hz.

Top to bottom: total impedance at 200°C, at room temperature, and the sensitivity change at the elevated temperature.

A solution
The SmartSub incorporates a solution that consists of measuring voice coil temperature in real time and using this information to adjust the gain and frequency response of the amplifier to correct the sensitivity and response changes that would otherwise occur. This correction ensures that the output is not compressed or imbalanced during high demand.

The SmartSub uses a heat sensor on the voice coil to measure voice coil temperature in real time to allow correction of sensitivity and response changes.

Wall and corner placement problems
It is well known that placing any speaker, including a subwoofer, near a wall will cause both an increase in bass level and unevenness in the response due to reflection from the wall. The effects of corner placement are more complex than single-wall placement since there are then two sets of low frequency level and reflection effects. In fact, these effects are quite predictable.

The general increase in level affects frequencies below approximately 300 Hz and its degree is dependent on the distance from the wall. It is caused by the fact that the sound radiation is confined to a smaller and smaller solid angle as the speaker's distance to the wall becomes less and less. This effect is illustrated in this graph for wall distances of 1.8, 1.0 and 0.5 meters.

Response changes caused by reinforcement from a nearby wall at distances of 1.8, 1 and 0.5 meters.

The reflection from the wall causes a partial cancellation in a particular frequency range which is determined by the distance to the wall. The severity of the cancellation is determined by the strength of the reflection which is also determined by the speaker-to-wall distance.

So, if the speaker is a relatively large distance from the wall, the cancellation will occur at a low frequency and be mild. If the speaker is close to the wall the effect occurs at a higher frequency range and is more severe. This second graph illustrates the wall reflection effect for the same three distances. When both these effects are added together, the result is illustrated in the third graph.

Response changes caused by reflection from a nearby wall at distances of 1.8, 1 and 0.5 meters.

Response changes caused by both reinforcement and reflection from a nearby wall at distances of 1.8, 1 and 0.5 meters.

Since all three variables of the boundary effect (degree of reinforcement, cancellation frequency and cancellation severity) are determined solely by the wall distance, they can be accurately compensated for if the wall distance is known.

All SmartSubs incorporate placement compensation whereby one control setting, calibrated in distance from the wall, corrects reinforcement, cancellation amount and cancellation frequency. By incorporating two sets of wall compensation which are independently adjustable, the SmartSubs are able to accurately compensate for virtually any placement.

For example, the fourth graph illustrates the effect of placing a speaker 0.5 and 1.2 meters from a corner. The inverse compensation supplied by the SmartSub for this placement is illustrated in the last graph.

Response change caused by near-corner placement of one-half meter from side wall and 1.2 meters from rear wall.

Correction supplied for a side wall distance of one-half meter and a rear wall distance of 1.2 meters.

There are two possible ways a subwoofer can be used to reproduce main channel bass: augment or crossover. Augment operation allows the main speakers to operate normally, without a crossover, and the subwoofer is used "fill-in" the deep bass below the range of the main speaker. Crossover mode transfers some of the bass range from the main speakers to the subwoofer. Each of these types of use can be used with two types of main loudspeakers, sealed or reflex. Following are examples illustrating typical results for each case.

It is assumed that the subwoofer crossover configuration is what I consider optimum for generic filters. The sub low-pass (LP) filter is 4th order, Q = 0.5, the main speaker high-pass (HP) filter is 2nd order, Q = 0.7, the HP frequency tracks the LP frequency, and there is also a continuously variable phase control.

The simplest case is to use the crossover with sealed main speakers. In the special case where the crossover frequency is set to the speaker's limit (-3 dB) frequency, the crossover's HP will combine with the speaker's bass response to achieve a total 4th-order, Q = 0.5 high-pass response that will, in theory, perfectly cross over with the 4th-order, Q = 0.5 sub low-pass response.

However, in practice, even this simple case is upset by the complication that the sub's response does not extend to DC. For an 80-Hz crossover, if the sub has a 4th-order rolloff at 35 Hz, the interactive phase effects will cause the whole bass range to be 2 dB weak (Fig. 1).

Fig. 1. Subwoofer extending to 35 Hz and crossing-over at 80 Hz with a sealed, 80-Hz main speaker. Standard settings.

Increasing the signal level to the sub solves this problem, but the system response still suffers from excessive output in the 100 to 200-Hz region.

In the more general case where the crossover frequency is higher than the main speaker's frequency, the results are not as good. For example, if the main speaker's response extends to 50 Hz and the crossover is at 80 Hz, the bass response is 4 dB weak (Fig. 2).

Fig. 2. Subwoofer extending to 35 Hz and crossing-over at 80 Hz with a sealed, 50-Hz main speaker. Standard settings.

Even when the phase and level are adjusted to be optimum (180°, -1 dB) the response is not nearly ideal (Fig. 3).

Fig. 3. Subwoofer extending to 35 Hz and crossing-over at 80 Hz with a sealed, 50-Hz main speaker. Optimum phase and level adjustment.

In cases where the main speakers are ported, things are no better. For example, if the main speakers are tuned to 50 Hz and you want to implement an 80-Hz crossover with the sub extending to 25 Hz, the results achieved without phase adjustment are not what is desired. Even when the phase control is set optimally (80° @ 50 Hz) the results are not good, giving a hump in the bass while still giving weak upper bass (Fig. 4).

Fig. 4. Subwoofer extending to 25 Hz and crossing-over at 80 Hz with a reflex, 50-Hz main speaker. Optimum phase and level adjustment.

If you want the sub to augment the main speakers, without adding a high pass into their response, the results are usually even less desirable. If the main speaker is a reflex type at 50 Hz and the sub extends to 30 Hz, the bass is very weak without a phase adjustment (Fig. 5); the polarity must be reversed to get good results.

Fig. 5. Subwoofer (attempted) extending to 30 Hz augmenting a 50-Hz reflex main speaker. No phase adjustment.

In the case of augmenting a sealed system, it is usually not possible to obtain good results. For example, a 50-Hz speaker will produce a severe null at the crossover frequency without a phase adjustment (Fig. 6).

Fig. 6. Subwoofer augmenting to 30 Hz a 50-Hz a sealed speaker. No phase adjustment.

Even with optimum phase and level, the results are an exaggerated mid bass and a lack of deep bass (Fig. 7).

Fig. 7. Subwoofer (attempted) extending to 30 Hz augmenting a 50-Hz sealed main speaker. Optimum phase adjustment.

These examples illustrate the typical problems inherent in matching a subwoofer to other speakers:

• "Logical" settings do not give the desired results
• It is not known which level and phase control settings will give optimum results. They must be adjusted without rhyme or reason by trail and error
• Optimum results are usually not very good.

Rather than telling the subwoofer how to perform by using controls for crossover frequency, level and phase, the THIEL subwoofer crossover unit has settings for the characteristics of the main speakers being matched, the configuration of the system and the performance desired by the user. This information is then used to calculate and provide the ideal subwoofer response and high-pass filter response that will give very accurate results with the user's main speaker.

There are two component techniques that are used to achieve these results.

1. Determining what LP filter characteristic for the subwoofer will perfectly match the HP characteristic of the main speaker, either HP-filtered in crossover mode, or not filtered in augment mode.
2. If a HP filter is supplied for the main speaker signal, determining what filter will, when combined with the response of the main speaker, provide a total response that perfectly matches the desired 4th-order, Q = 0.5 response.

Once the crossover unit knows the characteristics of the main speakers, it calculates and implements crossover rolloff shape, slope, phase and HP as needed to achieve perfect results. Not only does the crossover calculate and implement the optimum setting for the subwoofer cut point, slope and phase, but these characteristics are not limited to those of standard filter shapes and instead can be whatever filter shapes will give the desired results.

As an example of the first technique of generating subwoofer response that will blend correctly, Fig. 8 illustrates the subwoofer augmenting a 50-Hz reflex speaker, extending the bass response to 30 Hz. As is true in every case, the results in this case are essentially perfect. These excellent results are achieved because, in addition to the crossover automatically calculating the optimum phase response, the response shape of the subwoofer is exactly that shape needed to perfectly integrate.

Fig. 8. 50-Hz reflex main speaker response; SmartSub woofer response; net combined system response. Dotted is generic filter response.

For comparison, the dotted graph shows the response that a generic filter would provide. You can notice the somewhat lower output level around 30 Hz and a somewhat higher and more gentle rolloff slope. This "custom" slope is automatically generated and is exactly what is needed to match this main speaker.

Another example illustrates this "custom" slope ability of the crossover unit. If the sub is augmenting a sealed 60-Hz speaker, the crossover again provides the output that will produce exactly the desired combined output (Fig. 9). Again, the dotted line shows what a generic filter would provide and, in this case, the difference is large. The proper curve provides more output below 25 Hz and less above, with a slightly more gentle slope.

Fig. 9. 60-Hz sealed main speaker response; SmartSub woofer response; net combined system response. Dotted is generic woofer filter response.

The following example illustrates the second technique of providing a special HP response that will combine with the speaker's response so that the total response is correct. If an 80-Hz crossover is desired with a 60-Hz sealed main speaker, the HP supplied by the SmartSub crossover is an unusual shape that resembles an 80-Hz 4th-order above 60 Hz, and a 2nd order below (Fig. 10).

Fig. 10. 60-Hz sealed main speaker response; SmartSub crossover response supplied; net 80-Hz 4th-order Q 0.5 response.

If the same crossover is desired with a 60-Hz reflex speaker, you can see in Fig. 11 that the supplied HP shelves below 60 Hz so that, again, the response of the filter combined with that of the speaker will have the desired 80-Hz, 4th-order characteristics.

Fig. 11. 60-Hz reflex main speaker response; SmartSub crossover response supplied; net 80-Hz 4th-order Q 0.5 response.

About the author
Jim Thiel is a co-founder, co-owner, and product design engineer for THIEL. His interest in music began on the piano in early childhood and remains an important part of his life today. His involvement in audio can be traced to age 12 when he began to build and repair radios and other electronic gear. His academic background is in physics and mathematics. Jim pioneered the principle of time and phase accuracy in loudspeakers with the use of sloped baffles, coaxial driver mounting, and phase coherent crossover network design.

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