Loudspeaker Enclosure Design
John Kormylo - ELFSOFT

At low frequencies, where the physical dimensions are small compared to the wavelengths, one can treat the acoustic elements of a loudspeaker enclosure as a lumped parameter system. This model can be combined with the lumped parameter models of the electrical and mechanical components into an equivalent circuit model (see Appendix A). The net effect is that speakers act as second order high pass filters.

At high frequencies, the acoustics must be modeled as distributed parameter systems where shapes and internal reflections must be taken into account. These effects are largely undesirable and are minimized through the use of sound absorbing materials.

Individual speakers are only useful over a limited range of frequencies. Because of their high pass characteristic, speakers are ineffective below their lumped parameter resonance frequency. Aditionally, low frequencies require either greater transit distances or larger surface areas to produce sufficient sound volume. Directionality becomes a problem when the speaker diameter exceeds 1/2 wavelength. Voice coil impedance can reduce the gain at high frequencies, and the diaphragm has a greater tendency to flex.

The purpose of 2-way and 3-way speakers is to be able to reproduce sound accurately over as much of the audio band (20 Hz to 20 kHz) as possible.

The speaker is mounted into a wall between two infinite half-spaces (or at least two large rooms).

 electrical . mechanical . acoustic
            .            .
 V1         .     V2     .
 *--- Re -----+---+---+------+----+--- T(s) ---*
           .  |   |   |  .   |    |
           .  |   |   |  .   |    |
           .  Cm  Lm  Rm .  Z(s) Z(s)
           .  |   |   |  .   |    |
           .  |   |   |  .   |    |
 *------------+---+---+------+----+------------*
           .             .

Re is the DC electrical resistance (which can be measured directly), Cm is proportional to the piston mass, Lm is proportional to the mechanical compliance and Rm is inversely proportional to mechanical resistance. Z(s) and T(s) represent the load and coupling between the diaphragm and output sound for infinite half-spaces (see Appendix E). Z(s) applies to both sides of the speaker in this case. What is important is that Z(s) is very large and can be ignored, while T(s) is proportional to frequency, turning what would otherwise be a simple resonator into a high pass filter.

The transfer function and input impedance for this model (and others) are listed in Appendix B. A real world example is given in Example. <\p>

It should be noted that it is better damp the mechanical resonance by reducing Re than Rm, so long as one does this by increasing the wire diameter instead of reducing the length of the coil. Deliberately adding friction may be easy, but also reduces overall gain.

A simple speaker enclosure can be modeled as the following circuit:

 electrical . mechanical  . acoustic
            .             .
 V1         .      V2     .
 *--- Re ------+---+---+-----+---+---+--- T(s) ---*
            .  |   |   |  .  |   |   |
            .  |   |   |  .  |   |   |
            .  Cm  Lm  Rm .  La  Ra  Z(s)
            .  |   |   |  .  |   |   |
            .  |   |   |  .  |   |   |
 *-------------+---+---+-----+---+---+------------*
            .             .

Re, Cm, Lm and Rm are characteristics of the speaker itself. La is proportional to the volume of the enclosure and Ra is inversely proportional to the amount of sound absorbing material used. <\p>

Note, Lm and La can be replaced by a single inductor L = (Lm La) / (Lm + La), which is smaller than either L1 or L2 individually. R2 and R3 can also be replaced by a sincle resistor. Consequently, the analysis of the simple enclosure and the infinite baffle are almost identical.

Tupically one would like to have a maximally smooth (Butterworth) response with as low a resonance frequency as possible. Increasing the mass of the speaker diaphragm will lower the resonance frequency, but also reduces passband gain. Increasing the volume of the enclosure also will lower the resonance frequency, but will never reduce it below its unenclosed (infinite baffle) value. Adding sound absorbing materials will increase damping. In the unlikely event that the speaker is already overdamped, one could reduce the damping by adding a resistor in series with the speaker.

One can achieve any desired speaker response using electrical filters and overdriving the low frequencies to compensate for the limitations of the speaker itself. However, there are a few caveats.

Lower frequencies require longer traverses of the diaphragm to achieve the same velocity and therefore the same sound volume. A speaker diaphragm will only travel so far before it starts to hit things. A given speaker therefore has a natural limit on how much bass it can produce without distortion. Increasing the diaphragm size increases the amount of bass which can be produced, but requires a larger enclosure (L2 is divided by the diaphragm area squared).

Attempting to drive a speaker below its resonance frequency requires a lot of power, possibly exceeding the amplifier's capability or the speaker's ability to disipate heat.

Finally, high-power low-frequency passive filters require very large and expensive components. Using acoustics to achieve the same effect is generally much less expensive. Alternatively, active or digital filters are even less expensive, but require separate drivers for each speaker.

A bass reflex or vented speaker uses a "tuned port" to vent some of the sound from the enclosure. This has little effect on higher frequencies but causes an additional 6 dB/octave rolloff below the resonance frequency. The idea is to provide additional damping at the resonance frequency, especially since sound absorbing materials do not work well at lower frequencies.

The equivalent circuit for a bass reflex speaker is of the form:

 electrical . mechanical  . acoustic
            .             .
 V1         .      V2     .            V3
 *--- Re ------+---+---+----- Ca --+---+---+--- T(s) ---*
            .  |   |   |  .        |   |   |
            .  Cm  Lm  Rm .        La  Ra  Z(s)
            .  |   |   |  .        |   |   |
 *-------------+---+---+-----------+---+---+------------*
            .             .

Ca is proportional to the length of the tube divided by its area. (see Appendix A). The tube and enclosure cavity appear in series since they have a common pressure (through variable) and the total mass flow (across variable) equals the sum of the mass flows for the two components. The sound pressure at a distance is proportional to the mass flow from the speaker minus the mass flow from the port, which equals the mass flow across L2 (hense the placement of Z(s) and T(s)).

A more rigorous treatment would take into account two different ports while computing air loading. However, their effects over the frequencies of interest are negligable.

The transfer function and input impedance are listed in Appendix B. The traditional solution is to set Ca La = Cm Lm and adjust Ra to reduce the peak gain. Alternatively, one can produce a fourth order Butterworth response for specific values for Ra, La and Ca for a given speaker (Re, Rm, Lm and Cm).

I was curious what would happen if one subdivides the enclosure into two cavities connected by a tube. The idea was to use the resonator (L2 Ca) to boost frequencies below the rolloff frequency.

The equivalent circuit for this system is of the form:

 electrical . mechanical  . acoustic
            .             .
 V1         .      V2     .
 *--- Re ------+---+---+------+----+------+--- T(s) ---*
            .  |   |   |  .   |    |      |
            .  |   |   |  .   L1   Ra     |
            .  |   |   |  .   |    |      |
            .  Cm  Lm  Rm .   +----+ V3   Z(s)
            .  |   |   |  .   |    |      |
            .  |   |   |  .   L2   Ca     |
            .  |   |   |  .   |    |      |
 *-------------+---+---+------+----+------+------------*
            .             .

L1 and L2 are proportional to the volumes of the two cavities and Ca is proportional to the length/area of the tube (see Appendix A). Ra results from sound absorbing materials in the primary cavity (L1).

The transfer function and input impedance are listed in Appendix B. There is no maximally smooth solution for this system due to the presence of unremovable complex zeros in the transfer function.

If the main enclosure is larger than the resonator and the resonator is tuned below the speaker resonance, then the extra zeros will be below the pass region. The result would be to extend the pass region to lower frequencies, more so than by increasing the size of the main enclosure by the same amount.

Any enclosure has certain frequencies at which it resonates due to standing waves. Energy is stored up when the speaker drives those frequencies, then released again causing the speaker to "hum." Sound absorbing materials reduce the problem, but also reduce the overall efficiency of the system. Since their effect is to lower and broaden the "spikes" in the frequency response, they works better with many weak resonances than with a few strong ones.

For low frequencies, enclosures need a lot of volume, but shapes with good volume to surface area ratios have horrible resonance characteristics. Most enclosures consist of rectangular boxes which have good volume to surface area ratios and are packed with sound absorbing materials.

Ideally one would like an enclosures to have all of the standing wave resonances outside the pass band for the speaker. One such solution is to combine two short dimensions and one very long dimension (a waveguide or labyrinth). Shapes with sloped walls would do better, but enclosures with sloped walls can be treated as horns (see Appendix G) and it turns out that all horns still have resonance frequencies. The best shape I have come up with is to place two exponential horns mouth to mouth and use them below their critical frequency.

Any coil will possess impedance, although the effects of the permanent magnet will dominate at low frequencies. This impedance complicates the solution and is typically ignored during the design, especially since nothing can be done to compensate.

But as the frequency increases, this impedance inevitably decreased the effective gain of the speaker. Depending on the speaker parameters and the frequency range of interest, this could have serious consequences and may need to be included in the model.