Loudspeakers 619The trapezoid designation describes the plan view of the
enclosure, and the shape allows multiple loudspeakers
to be arrayed in the shape of an arc segment, with the
included angle between adjacent loudspeakers being
equal to twice the sidewall angle. A large number of
manufacturers offer loudspeakers that fit this descrip-
tion. Typical sidewall angles range from 12° to 15°,
while the horizontal coverage angle (the angle at which
the output has fallen to 6 dB below the level on axis) of
the high-frequency horns used in such devices is typi-
cally either 60° or 90°, and the coverage of the woofer
is entirely uncontrolled.
17.8.3 Performance Issues in Multiway Systems
Before trying one’s hand at designing a multiway loud-
speaker, it is a good idea to develop a familiarity with
the complete audio signal chain and to understand the
implications each decision will have on the acoustic
signal that will reach the listener’s ear. Fig. 17-42 is a
functional diagram of the electrical and acoustic signal
paths from the source (electrical input signal) to the
observation (listening) point.
In the above representation,(17-6)
where,
FT is the total electroacoustic transfer function,
Fn is the (electrical) transfer function of the nth cross-
over filter,
S is the Laplace complex frequency variable,
Gn is electroacoustic transfer function of the nth radiator
in the system,
x, y, and z are Cartesian spatial coordinates.The concept is general and will accommodate an arbi-
trary number of spectral divisions.
It should be noted that, although this diagram depicts
a loudspeaker with a passive crossover, it might also be
used to represent an active system simply by including
gain in the transfer functions of the crossover blocks.
An active loudspeaker simplifies the task of crossover
design. Since the power amplifier serves as a buffer
between the crossover and the transducers, the
frequency-dependent impedance behavior of the trans-
ducers becomes a very small factor in the design
process. Note that the crossover filters are in cascade
with the transducers, while the acoustic outputs of the
devices are summed acoustically at the listening posi-
tion. The nature of such a multipath system is complex,
and detailed prediction of its response at every likely
listening position is a nontrivial task.
Note also that the transfer functions Gn are functions
of the spatial coordinates x, y, and z as well as of the
complex frequency variable S. This spatial dependency
includes the effects of source directivity, propagation
delay, and the inverse square law. If the system is not
coaxial (and sometimes even if it is), then the lengths of
the paths from each transducer to a given listening posi-
tion will not generally be the same. The most common
practice is to choose an axis along which one will
attempt to equalize these acoustic path lengths and then
to optimize the speaker’s behavior on this axis. In the
case of a two-way loudspeaker with the transducers
displaced along a line in the plane of the baffle, it is
possible to make path lengths equal at every point in a
plane. Once more than two spectral divisions are present,
even this limited goal is no longer possible. It may well
be the case that, in a three- or four-way system, there is
no point for which all acoustic path lengths from the
transducers to a listener’s ears will be equal.
The effect of unequal acoustic path lengths is that
signals from different radiators will reach the listener at
different times, even though they originated simultane-
ously. This timing discrepancy is not generally suffi-Figure 17-41. Two-way monitor speaker. Courtesy Genelec.
Figure 17-42. Functional diagram of multiway loudspeaker.
F 1 (S)F 2 (S)F 3 (S)G 1
(S, x, y, z)G 2
(S, x, y, z)G 3
(S, x, y, z)3FT 6 Sxyz = N>@Fn SGn Sxyz