Since Rob asked me to chime in, I'll add a couple of thoughts to the discussion.
There is a terrible impedance mismatch between a normal direct radiator cone speaker and the air it is trying to move... sort of like swinging your fist at the air. The weight of the cone is much heavier than the air. It is very difficult to transfer energy (accomplish work) this way, and very low efficiency results. If we couple the speaker to a horn with a mouth much larger than the throat and an adequate length, now it has the trapped volume of air in the horn to push against, and it can create sound energy much more efficiently. Many benefits ensue- driver excursions are greatly reduced for a given sound output, and those distortions proportional in some way to excursion (most of them) are reduced greatly as well. For a given input power, sound output power is increased, often by an order of magnitude or more. This both increases the maximum output capability and decreases the power input for a given volume level.
A compression driver is a specialized type of speaker designed to exploit the benefits of horn loading to the maximum. Imagine removing the cone driver from the horn in the previous example, and lengthening the horn on the throat end down to a very small throat area. Now we mount the compression driver, whose internal sound channels continue the reduction in cross section right to the tiny slits in the surface of the phasing plug. Now instead of a relatively soft, heavy paper cone, we can use a formed metal or composite diaphragm, super light, thin, and shaped for maximum rigidity. The working surface is likely ten times greater than the area of the slits. Since the diaphragm must push air in and out of these slits and is loaded by the entire mass of air in the horn, it is easy to see that the pressure created by a given movement of the diaphragm will be enormous compared to a cone speaker driving the air directly. This increase in loading, or radiation resistance, allows for an efficiency (sound power output compared to electrical power input) of around 35 percent. Most direct radiator speakers operate at one percent efficiency or less (some high efficiency designs like JBLs can manage two or three percent). That's right- one percent sound, 99 percent heat. It takes quite a motor to drive a compression driver diaphragm, and everything is done to make it as strong as possible. Tolerences are tight, as a narrow gap is necessary to maintain very high flux density, often more than 20 kilogauss. The voice coil is most often edgewound, to maximize conductor length in the available space.
We usually think of compression drivers as midrange or high frequency devices, but they can be used at bass frequencies as well, with the same benefits. The Fletcher System built by Bell Labs in 1933 used a low frequency compression driver with an 18" spun aluminum diaphragm, 8" edgewound aluminum voice coil, coupled to a large reentrant bass horn. Usually, paper cone drivers are used on bass horns with large throat areas, as it allows considerable shortening of the horn.
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