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Topic No. 4

Loudspeaker Design Tradeoffs
John L. Murphy
Physicist/Audio Engineer


Loudspeakers have very definite performance limitations.

If a designer understands those limits he will be in a better position to "fit" a particular speaker design to the application at hand. To get some perspective on the matter, let's start with an exaggerated example of what we cannot do with a loudspeaker: we can't make a speaker that goes very low, is highly efficient, and uses a tiny enclosure.

Imagine a speaker that went to 20 Hz with a sensitivity of 100 dB SPL in a .25 cubic foot enclosure. It is by no means obvious but such a speaker cannot be made using today's technology. The loudspeaker analysis developed by Thiele and Small (so called, "Australian Loudspeaker Theory" because both of these men were from Australia) shows that there is a definite relationship between low frequency bandwidth, efficiency, and box size. I like to compare the situation to a pie which can be divided into three parts representing: bandwidth, efficiency, and compactness. The size of the pie is fixed but you can divide the pie however you wish. If you want a bigger slice of efficiency then the bandwidth or compactness slices will have to be smaller (meaning less bandwidth and/or a larger box). If you want extended bass response then either the efficiency or the box will have to give. If you want a tiny enclosure then bandwidth or efficiency will have to be sacrificed.

Loudspeaker Tradeoff Factors

Traditionally, designs have played out something like this:

For sound reinforcement applications we emphasize efficiency at the expense of bandwidth and box size. That is, sound reinforcement speakers tend to be efficient but with somewhat limited low end bandwidth and using large boxes. A typical 3-way box might have the following characteristics:

Sound Reinforcement Speaker
Sensitivity: 100 dB SPL (1W/1m)
Bandwidth: -3 dB at 60 Hz
Enclosure: 3 cubic foot

For Hi-Fi applications the emphasis is more on low frequency extension and box size. A typical Hi-Fi speaker might have the following characteristics:

Hi-Fi Speaker
Sensitivity: 90 dB SPL (1W/1m)
Bandwidth: -3 dB at 40 Hz
Enclosure: 1 cubic foot

A Mini Hi-Fi speaker places the emphasis on size and bandwidth and might have the following characteristics:

Mini Hi-Fi Speaker
Sensitivity: 80 dB SPL (1W/1m)
Bandwidth: -3 dB at 70 Hz
Enclosure: .25 cubic foot

These are just approximate numbers to give you an idea of how the tradeoffs go. In any event the following specifications are NOT physically possible to achieve under Australian Loudspeaker Theory:

Impossible Speaker!
Sensitivity: 100 dB SPL (1W/1m)
Bandwidth: -3 dB at 20 Hz
Enclosure: .25 cubic foot

Knowing these tradeoffs, a speaker designer is more likely to come up with a good compromise (i.e. pie division) when his customer asks for a PA speaker that goes to 20Hz, at 110 dB SPL and fits in his briefcase!

The following mathematical relationship between these parameters can be found in Small's paper :

Closed-Box Loudspeaker Systems
published in the Journal of the Audio Engineering Society
Vol. 20 No. 10 (DEC 1972)

Efficiency = k * f(3)^3 * V(B)
(where k is the so called "efficiency constant")
(f(3) is in Hz, and V(B) is in cubic meters)

for closed boxes: k = 2 * 10^-6

for vented boxes: k = 4 * 10^-6

Note that these are maximum values for the efficiency constants for the two types of boxes. The actual value depends on the specific alignment of the system.

Notice the EXTREME sensitivity to f(3) in this relationship (because it's cubed). A little bit more bass extension costs you a lot of efficiency or box size. Yet box volume and efficiency are traded equally. If you want to reduce the size of the box by half, then efficiency will be halved also provided the f(3) stays the same.

If we study the efficiency equation we can see that in order to achieve one octave of bass extension requires a payment of a factor of 8 in efficiency or box size. For example, say you have a system that has a F(3) of 80 Hz and you want to redesign it so that it goes to 40Hz.You could either make the box 8 times larger or have the system end up 1/8th as efficient. Or you could make the box twice as large and end up with 1/4th the efficiency. Either way, you must PAY mother nature for the extra octave of bass. There can be no cheating.

It is also important to realize that for typical closed and vented box systems the efficiency of the system is totally determined by the choice of driver. Once a driver is selected you can vary the enclosure to change the bandwidth but the efficiency will not be affected by the choice of enclosure. So driver selection is VERY important in the overall design procedure. In other words, don't expect to get a lot of low bass from a super efficient pro sound driver. On the other hand, don't expect to build stage monitors using low efficiency hi-fi drivers.

Hopefully this shows the role of the speaker designer as one of compromise. Now, your job is to explain this to your customer who wants the briefcase speaker for concert sound!



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