My Approach to Bass Horn Design

Many starting points for bass horns are built by the equations given in a paper by the late Marshall Leach [1]. These equations are built on certain assumptions, and will give you either a horn system based on driver parameters, or driver parameters and system based on a set of specifications. These assumptions are:

The impedance match is designed for maximum sensitivity
The low frequency corner is set by the rear volume
The horn cutoff is set at the same frequency, and the T-value of the Hypex horn selected to provide reactance annulling with the specified rear volume
The upper corner frequency is set by the mass of the driver
The front chamber is used to extend the HF response slightly before it starts to roll of quicker

I don’t like Leach’s approach to reactance annulling. For some reason he believes that it is the rear chamber that should set the LF -3dB point, presumably this is in order to preserve space, as it sort of sets the minimum chamber volume for the specified LF extension. But by doing this he paints himself into a corner: Reactance annulling using a plain exponential horn becomes impossible, and he is forced to use a Hypex horn, with a T-value that often ends up around 0.5.

Another thing I don’t like with this approach is the way reactance annulling is considered to just equating the horn reactance with the system compliance at the cutoff frequency, essentially resonating the horn air mass with the system compliance. This approach comes from Plach and Williams [2], who combined the concept of reactance annulling, originally invented by Albert Thuras, with the Hypex horn. With an exponential horn, reactance annulling works perfectly, since the shape of the horn reactance curve perfectly matches, or conjugates, (in the case of an infinite horn) the reactance of a compliance. The system reactance is therefore effectively nulled out above horn cutoff, increasing power output. With a Hypex horn, horn reactance is less above cutoff, but greater near cutoff. The lower the T-value, the worse the reactance annulling above cutoff. This is clearly illustrated by the fact that the total system compliance becomes zero with T = 0.

The original Bell Labs approach to reactance annulling was not to resonate the horn reactance with the compliance, but to create a conjugate impedance match between driver and horn. This is possible to do with Hypex horns too, which I have shown in a paper.

A More Basic View

Instead of just going by a set of formulas form a paper, it can be useful to instead look at some basic horn-driver interaction. For a bass horn, there are three important relations:

The midband impedance match between driver and horn. This sets the efficiency, sensitivity and, since the product of efficiency and bandwidth (EBP) is constant, also the bandwidth of the system. In the midband, the horn can be considered a resistive load having a value of
while the driver presents a source impedance of
These two impedances should be equal for maximum sensitivity, while maximum efficiency requires a maximum throat area. In many cases, we just equate these expressions, solve for throat area S_t, and are done. But we don’t have do go about it that way, as we will see.
The total system compliance relative to the horn reactance. This has to do with low frequency extension, driver protection from subsonic diaphragm excursion (since the horn doesn’t load the driver below horn cutoff), and efficiency above cutoff due to reactance annulling.
The diaphragm mass relative to the horn resistance R_AL. This sets the upper -3dB corner frequency of the power response. But since the horn has typically started to beam in that frequency range, unless the mouth is very small, the on-axis response doesn’t roll off at this point. But we should still pay attention to the mass corner frequency if we want to use the horn in the resistance controlled frequency range, where the driver is truly horn loaded. If we want to roll off the power response, we can add a front chamber. But this makes the power response roll off at 12dB/octave, and it may be harder to cross over. One reason to add a front chamber, though, may be to filter out nastiness from the driver, like distortion products and cone breakup, but this should hopefully not be a problem in a domestic setting.

The importance of the impedance matching described by relations 1 and 2 is that it reduces the variations in power output (and therefore creates a smoother response) when the acoustic load varies. This allows us to use a smaller mouth area than would otherwise be required, but more importantly, it helps in dealing with room modes. A bass horn in a small room will operate in the modal region of the room, with large fluctuations in radiation impedance. Even when placed in a corner, the throat impedance will be ugly. Thuras has show that the variation is smallest when

i.e. the geometric mean of the minimum and maximum throat resistance values in the working range. This factor is usually close to 1, so R_AL = R_AT is a useful starting point.

The relations also show us that we have many levers to pull in order to achieve an impedance match: compression ratio Sd/St, driver motor system parametersBl and R_E, and amplifier output impedance R_G.

Driver

In many cases we have selected a driver to design a horn around, based perhaps on a figure of merit (EBP, Bl, etc), word-of-mouth or availability. I have a pair of Altec 515-8G, which are great bass horn drivers (it actually says so on the back), but I also work in a loudspeaker factory, so I could design and build a suitable driver for my horn. There is also the option of calculating some parameters and then look for a suitable driver that matches those parameters. Let’s have a look at the impedance matching relations.

Ignoring R_MS,

and we see that a driver with a strong motor results in a small throat. We may increase the throat area by using a weaker motor, larger diaphragm, or higher amplifier output impedance. So using a tube amplifier is not out of question either, if we play our cards right.

Rearranging the equation to give us Bl, we get

Another factor in choosing throat area is horn distortion. A smaller throat means more power per area, i.e. higher intensity in the throat, and this leads to higher pressures and higher distortion. So we don’t want to make the throat area too small. For domestic use, however, horn distortion is not a major concern.

When it comes to driver mass and compliance, this is linked to driver f_s and Q values. But these compound parameters obscure the relations between the horn and the driver parameters, and can make the whole thing more confusing than need be. For instance, the mass corner frequency is set by the system resistance and moving mass:

But if the various electromechanical parameters are expressed by Thiele/Small parameters, and we assume impedance matching and that all system compliance is provided by C_MS, this equation can be rewritten as

The relation

is called EBP, efficiency-bandwidth product, as it is the product of the mass corner frequency and the conversion efficiency, and is a driver figure of merit. But EBP may put too much emphasis on the raw driver having a high resonance frequency, since the influence of the rear chamber is left out of the picture: the lower corner frequency is left out of the picture. Actually, with a soft suspension and a small rear chamber, a low resonance frequency is perfectly acceptable, as long as M_MD gives the correct mass corner frequency. This is one reason I don’t like T/S parameters for horn design: their use rely on assumptions that are not satisfied in horn speakers. They were originally devised to aid in the design of direct radiator speakers, where the system can easily be represented by a few lumped elements. Additionally, they lump the air load mass in with the driver parameters, which are allowable in direct radiators where you know exactly the acoustic load the driver will face, but it is not in horn design. T/S parameters can be misleading in this case.

Driver compliance (C_MS or V_AS) work together with the rear chamber to provide the system compliance for reactance annulling. The exact value is insignificant, as long as it is large enough to not require an overly large (or negative) cabinet volume. It may be advantageous to make the rear chamber provide much of the compliance rather than a nonlinear suspension, although this may be a problem in high power design where the cabinet compliance becomes to stiff at high levels. With the low displacement possible with high efficiency horn speakers, the chamber compliance should be very linear. Looking at the driver C_MS(x) (or K_MS(x)) curves, one will often look for linearity and symmetry, but it should be kept in mind that it is the total compliance, including the nonlinear box, that should be evaluated, not the raw driver.

The Design

My preferred way of going about a bass horn design and driver selection is to design the horn first, based on a set of choices dictated by the acoustical performance of the horn. This approach is for a traditional wide-band bass horn, which is designed to be the only low frequency channel of the system, covering from the lowest frequency of interest (usually determined by how big a horn you can accept) up to 3-400Hz or so. So not a horn subwoofer, which is quite narrowband and can have a small mouth and be aggressively folded, nor a midbass horn designed for better MF extension and LF extension usually not below 80-100Hz.

A mouth large enough to reduce reflections to an acceptable level, creating a well-behaved and not too resonant throat impedance.
Set throat area based on distortion, and also considering the size of the driver and suitable compression ratio. A too high compression ratio may put too much stress on a thin, light cone, so a ratio in the range 2-4 is fairly safe.
Use an exponential horn. The reason for this is that if the mouth isn’t large enough, a Hypex horn (T<1) will have larger impedance peaks than an exponential horn. Also reactance annulling becomes more challenging.
Select a driver based on the desired impedance matching ratio, usually as good a match as possible. This determines driver (Bl)² ⁄ R_E.
Further the selection of drivers is narrowed down by looking at the mass corner frequency and resulting rear chamber size.

References

[1] Leach, W. M.: “On the Specification of Moving Coil Drivers for Low-Frequency Horn-Loaded Loudspeakers”, J. Audio Eng. Soc. , Vol. 27, 1979, No. 12 p. 950-959

[2] Plach, D. J.: “Design Factors In Horn-Type Speakers”, J. Audio Eng. Soc. , Vol. 1, 1953, No. 4 p. 276-281