Author: Roy F. Allison [President, Allison Acoustics, Inc. Natick, Mass.
01760, USA]
The work of Thiele' and Small in recent years has changed the procedure
by which loudspeaker systems are designed. They did not extend the inherent
limits of performance of low frequency loudspeakers, but they did define
those limits. They quantified the interrelationships among cabinet size,
system efficiency, and lowfrequency response range, and they developed
systematic synthesis techniques which enormously simplified design procedure.
It is certainly true that this has resulted in an improvement in the average
quality of loudspeaker systems, because it is now easier for manufacturers
to optimize their designs.
Indeed, several manufacturers emphasize in advertisements that their products
have been designed by computers programmed with the Thiele/Small synthesis
formulas.
Unfortunately, consumers can enjoy the benefits of this improvement only
to a very limited degree. It is traditional to design loudspeakers for
flat frequency response in an anechoic test chamber. The Thiele/Small design
formulas assume that kind of reflectionfree environment. But people do
not use loudspeakers in anechoic chambers; they use them in domestic living
rooms. Normal rooms react on loudspeakers in the bass range, and this can
change their power output significantly.
We can understand why this is so if we look at the formula for reference
sound pressure from a loudspeaker at low frequencies.
This is derived by Beranek [1] as follows:
(1)
where Uc = volume velocity of diaphragm; eg= opencircuit amplifier
voltage; B= magnetic flux density in air gap; l = length of voicecoil
wire in air gap; So = area of the diaphragm; Rg= output impedance (resistive
component) of the amplifier; RE= voice coil wire resistance; RA = sum of
electrical, mechanical, and acoustic resistive elements; ω = 2 pi f radian
frequency; MA = sum of the acoustic masses, including diaphragm and voice
coil, and CA= total acoustic compliance of suspension and box. All elements
are transformed to acoustic impedances and expressed in mks units.
Above the resonance frequency but within the piston band, and on the reasonable
assumption that RA^{2} is small relative to w^{2}MA^{2} (1) can be simplified to
and this is called the reference volume velocity.
Also, a loudspeaker system in this frequency region is a small source,
that is, its radiation is essentially omnidirectional. For such a source
the sound pressure is related to the volume velocity by
(3)
where pc= sound pressure at a distance r from the loudspeaker, f= frequency
in Hertz, and pc, = density of air. If we combine (2) and (3), we obtain
the formula for reference sound pressure:
Most of the terms in this formula do not change with the loudspeaker system's
environment, but two of them do change. One is the numerical factor, 4n,
in the denominator. This denotes the solid angle (the threedimensional
space) into which the woofer radiates sound energy. The solid angle is
measured in steradians; 4n steradians represents full space, which is the
radiation angle that would be seen by the woofer if the system were suspended
in midair high above the ground. Because lowfrequency sound is radiated
omnidirectionally, the woofer in such an environment would propagate sound
equally in all directions and the sound energy would continue outward without
encountering any nearby reflecting boundaries.
An anechoic chamber simulates such an environment.
Now if the loudspeaker system were to be mounted in the middle of a very
large wall, so that the woofer diaphragm were flush with the surface of
this boundary, the radiation angle would be halved. The woofer would be
operating in a halfspace environment, a solid angle of 2pi steradians.
The radiation formula tells us that the reference sound pressure will
be doubled, and experiment shows this to be true. We do find a 6dB increase.
An increase of 3 dB is obtained because we have confined all the radiated
power to half the space it occupied before. This 3 dB is of no importance
because it involves no change in power output. But a second increase of
3 dB is the result of an actual doubling of power radiated by the woofer.
It is twice as efficient in 2pi space as it is in 4n space.
Let us further suppose that we are able to position the woofer at the
intersection of two mutually perpendicular walls. Again the radiation angle
is halved, this time to pi steradians; again the power output of the woofer
is doubled.
Still another reduction in radiation angle, to pi/2 steradians, is often
found in real living rooms when a loudspeaker system is placed in a corner
with the woofer close to the floor. Its power output is increased another
3 dB at low frequencies, for a total increase of 9 dB above the power radiated
into full space.
It should be emphasized that these theoretical increases in radiated power
with decreasing radiation angle are obtained only when the driver is close
to the boundaries. In an acoustical sense, drivers are never "close" to
room surfaces at frequencies above 500 Hz or so. Therefore, real rooms
by changing a loudspeaker system's power output in one frequency region
only, in an amount determined by where the user places the cabinet in the
room completely alter the original balance of the system as it was in the
anechoic chamber. And because such room effects cannot be avoided, it follows
that loudspeaker systems designed for flat response in. anechoic chambers
cannot produce a flat power response in real rooms.
We have said that a driver must be close to a room boundary in order to
be affected by it. A discussion of what "close" means brings
us to the other term in our formula which changes when the loudspeaker
system is brought out of the anechoic chamber and into the listening room.
That term is' p0, the air density.
Of course, the average density varies only in accordance with changes
in barometric pressure. But instantaneous density is proportional to instantaneous
pressure, and sound itself is the result of variations in air pressure
which occur at an audible rate. In the anechoic chamber, a loudspeaker
driver creates audiofrequency changes in pressure which travel to the
chamber walls and are absorbed there. In a living room, conversely., sound
energy reaching the walls is reflected back to the driver where it either
adds to or subtracts from the instantaneous pressure on the driver diaphragm,
depending on the phase relationships. If a boundary is "close," the
reflected pressure is strong enough to change the instantaneous pressure
at the diaphragm surface significantly and to increase or decrease the
acoustic power radiated.
Closeness of a driver to a boundary is. dependent not on the absolute
distance between them, but upon the distance in units of wave length.
Therefore, a driver may be acoustically close to a boundary at low frequencies,
for which sound waves are long, and not close at higher frequencies which
have shorter wave lengths.
This relationship is quantified in Figs. 1 through 3. The power output
of a directradiator driver in its nondirective frequency range is shown
(relative to its output in an anechoic chamber)
when it is put near a single boundary (Fig. 1); when it is equidistant
from two boundaries intersecting at a right angle (Fig. 2), and when it
is equidistant from three mutually perpendicular boundaries (Fig. 3). In
each case the horizontal scale is in fractional parts of a wave length.
Figs. 13 Power output of small acoustic source close to one, two, and
three boundaries relative to power output in 4 pi space.
Fig. 4 Output of a typical highquality loudspeaker system in anechoic
test chamber (dashed line), and power output of same system with woofer
50 cm from three mutually perpendicular room surfaces ( solid line).
Fig. 5 Power output of same system as Fig. 4, but with woofer located
20 cm above floor, 51 cm from one wall, and 122 cm from other nearest wall.
We can learn much from a careful examination of these curves. For example,
it is obvious that the augmentations of power output predicted for reductions
in radiation angle (+3, +6, and +9 dB for 2n, Tr, and pi /2 steradians,
respectively) are obtained only when the woofer is placed a small fraction
of a wave length from the boundaries. At a distance of 1/10 wave length,
the threeboundary augmentation has already decreased from 9 dB to 7 dB.
A typical distance of the woofer from the three nearest walls in a real
living room might be 50 cm, and 50 cm is 1/10 wave length at 69 Hz. At
higher frequencies, the augmentation decreases rapidly, because 50 cm becomes
a larger fraction of a wave length as frequency increases.
Eventually, as we continue to increase the frequency of the sound signal,
the reflected pressure from the nearby walls no longer arrives back at
the woofer diaphragm in phase with the driving signal, and at that frequency
or above it the driver is no more efficient than it is in full 4 pi steradian
space. This limiting frequency is that for which the distance to the walls
is 1/4 to 1/5 wave length, depending on the number of boundaries involved.
In our hypothetical example, with a woofer located 50 cm from each surface
in a threeboundary intersection, that frequency is approximately 140 Hz.
At still higher frequencies (200 Hz in our example) the woofer is about
3/10 wave length from the walls. Then the reflected pressure is in opposite
phase with the motion of the diaphragm. As the woofer attempts to create
a compression, the reflections cause a rarefaction, and vice versa. The
woofer believes that it is working in a partial vacuum; its power output
is correspondingly reduced. But the amount of power reduction is not in
neat alignment with the number of reflecting boundaries. As Fig. 1 shows,
an outof phase reflection from one boundary has a minimal effect, with
only 1dB reduction at the worst frequency.
Equidistant reflections from two boundaries are considerably more effective,
causing a 3dB drop below fullspace output in the critical frequency band.
But reflections from three equidistant boundaries cause a deep reduction,
more than 11 dB, below reference level. The total variation (including
the 9dB boost at very low frequencies) is about 20 dB from a speaker system
that measured flat in an anechoic chamber! Figure 4 shows the power output
vs. frequency for such a system, assuming a system resonance frequency
of 50 Hz with a Q of 1 and 50cm distance to each of three room surfaces.
The curves also show that, beyond 5/10 wave length distance, the boundaries
have insignificant effect on loudspeaker performance. In our example, a
50cm distance is 5/10 wave length at 345 Hz.
Figures 1 through 3 make it apparent that the effect of multiple outof
phase reflections is more than simply additive. It follows that one should
attempt to place the woofer of a conventional speaker system so that its
distances to nearby room surfaces are as different as possible, and thereby
place the outofphase reflections at widely different frequencies. When
this can be done, it is very helpful in smoothing the system's lowfrequency
output. Power output vs. frequency is shown in Fig. 5 for our reference
system when the woofer is placed 20 cm above the floor, 51 cm from one
wall, and 122 cm from the other nearest wall.
Another way to deal with the effects of room boundaries is to place the
woofer far away from them all. If there is at least 1.25 meters from the
woofer to the nearest room boundary, all the major effects of the boundaries
will occur well below 100 Hz, where they will be less audible than they
are normally. Of course, this would not be convenient in most living rooms.
Both of these stratagems are defensive maneuvers. They limit the damage
but do not prevent it entirely. A frontal attack is preferable in this
case. It consists of designing loudspeaker systems to work in conjunction
with the room boundaries, taking full advantage of their ability to increase
efficiency at low frequencies and also avoiding the outofphase reflection
problem altogether. These are several ways in which this plan can be implemented,
but all have these steps in common:
1. Decide whether the system is to be used in .proximity with one, two,
or three mutually perpendicular room surfaces. The balance must be set
differently for each kind of system because, as we have seen, the woofer's
efficiency changes with its radiation angle.
2. Design the cabinet to locate the woofer ( or woofers) as close as possible
to the room surfaces.
3. Set the crossover frequency so that the operating range of the woofer
is limited to well below the "notch" frequency. Then the reflected
pressure will always be in phase with, and will reinforce, the direct output
of the woofer. The woofer's power output can then be flat. Note that in
practice this requires a threeway or fourway loudspeaker system, because
even with placement of the woofers very close to the boundaries a low crossover
frequency (400 Hz or lower) must be used.
4. Finally, locate the midrange driver in the cabinet far enough away
from the boundary intersection so that it is at least onehalf wave length
distant at the crossover frequency and above.
Then the output of the midrange driver (and of the tweeter) can be made
flat and unaffected by the room surfaces.
Some loudspeaker systems designed in accordance with these principles
are now available. I feel certain that others will soon follow.
Reference
1. L. L. Beranek, Acoustics (McGrawHill, New York, 1954), chapter 8.
(Source: Audio magazine, Aug. 1979)
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