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Diminishing Returns: When “A Little More” Isn’t Enough For A System

The principle of proportional change is the very heart of the decibel system

Intuition is one of the most valuable tools of the sound system designer. But sometimes intuition can lead us astray. Humans tend to think in terms of linear relationships rather than proportional relationships.

But most significant changes in an audio system are related to the proportional change rather than linear change.

For example, a 2-to-l power ratio produces a 3 dB increase in level, regardless of the initial power. A 1 watt to 2 watt increase produces the same proportional change as a 500 watt to 1000 watt increase – not exactly intuitive. This is why audio relationships are normally plotted on logarithmic scales.

Also interesting is the fact that it usually requires a 2-to-l change to make a significant difference when human perception is considered.

Most have heard the riddle of the “doubling penny.” A ruler offered a man the choice between a $1 million lump sum payment or a single penny that would be doubled each day for one month. The wise man took the penny realizing that the 2-to-1 growth rate would make him far wealthier than a mere one million dollars.

In fact, there would quickly come a time where it would be no longer possible to double the sum with each successive day, because the man would already possess all of the wealth of the kingdom!

Audio system designers must understand when they have reached the point of diminishing returns – the point when it is no longer practical (or possible) to double the parameter being considered. Let’s look at a few examples.

Amplifier Power

One electrical watt makes a handy reference quantity when considering what size amplifier to buy. This relatively small amount of power can only be practically doubled about 10 times before one reaches the point of diminishing returns.

In other words, the first few 3 dB level increases are quite economical. Then it becomes increasingly more difficult and expensive to get the next 3 dB. This makes amplifier peakroom pretty expensive, since it must be added after the power required to produce the perceived loudness is determined.

Each 3 dB reduction in available peakroom reduces the amplifier size by one-half. Peak limiters are an important part of the signal chain!

System designers must have other ways of achieving the desired level. And these other methods will also reach a point of diminishing return. One of these ways is increased loudspeaker directivity.

Directivity Factor

An omnidirectional loudspeaker radiates sound in all directions. It is said to have a directivity factor (or Q) of one.

If the sound radiation is limited to one-half of a sphere, then the directivity factor becomes 2 and the broadband sound level increases by 3 dB.

The next 3 dB requires that the directivity factor be increased to 4, and so on. After about five doublings, the system designer reaches the point of diminishing returns.

Once the Q is 32, it must be increased to 64 to get another 3 dB of gain. And once at 64 it must increased to 128. This is one reason why medium-Q (Q<= 20) loudspeakers dominate the marketplace. They mark the point of diminishing returns for achieving gain through directivity at midrange frequencies. Also consider that since directivity is related to wavelength, a horn diameter must double in size to extend its low frequency control by one octave. This makes horns that extend below 250 Hz impractical for some applications. We've reached the point of diminishing returns.

Direct & Reverberant

Another way to increase the direct field level is through the use of multiple loudspeakers. Adding a second loudspeaker provides a 3 dB broadband level increase. But the next 3 dB increase requires four loudspeakers, and the next, eight. Soon we have more loudspeakers than can be practically transported.

The point of diminishing returns principal can also work for the system designer. In reverberant spaces, adding a second identical (and identically powered) loudspeaker to cover some remote listeners increases the reverberant field level by 3 dB, often hindering communication in the space. It takes two more to drive the reverberant level up another 3 dB, and four more for the next 3 dB.

Eventually, it simply doesn’t make much difference to the reverberant field level when more loudspeakers are added. Once you have 100, you can add 100 more with only a small reverberant level increase.

Level and Distance

The inverse square law states that the sound level will drop by 6 dB when the distance from a point source is doubled in a free field. This means that sound level will fall much more quickly near the source than it will at remote distances. This makes it difficult to contain the sound produced at outdoor venues, which is often a source of aggravation to neighbors.

It also makes it difficult to achieve even sound coverage of large audiences from ear level loudspeaker placements of “point-source like” devices. Overhead placements equalize the distance differentials to various seating positions. The desire of many designers to produce an acoustic image near the talker has resulted in many poorly performing systems.

Damping Factor

Loudspeaker damping can be improved by reducing the resistance of the speaker wire, but only to a limited extent. There quickly comes a point when it is no longer practical to halve the resistance by increasing wire diameter or by shortening its length.

When a user realizes that a proportional change is required to improve performance, they will quit wasting time and money trying to take their speaker wires past the point of diminishing returns.

Other Areas

Of course proportional relationships are not limited to sound radiation. Light behaves in similar fashion. Keep that in mind the next time you shop for video projectors.

A doubling of lumens is required for a significant change, so upgrading from 800 to 1000 lumens is likely to be a disappointment. In the computer world, processor speed and system RAM are also perceived proportionally. Changes in either that are less than double the initial amount are probably not good investments. (Other than psychologically!)

Our current technology will reach a point of diminishing returns within a few years, limited by the speed of electricity across circuit boards. The same would be true when comparing the horsepower ratings when you shop for your next automobile, or the BTU ratings of space heaters.

Proportional changes are what makes our investments grow with time, and why the “time” in the stock market is what provides the return on investment. (Or at least it used to.) Those who start investing at a young age with little money can end up with much more than those who invest heavily at a later time.

The principle of proportional change is the very heart of the decibel system.

System designers that are not using the decibel are often responsible for large expenditures that produce little benefit, not realizing that they have reached the point of diminishing return with some aspect of their design.

The prudent designer will realize when a different path will produce a more significant result.

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