Handbook for Sound Engineers

Loudspeakers 623

1. Crossover frequency: below this frequency, output
   from the low-frequency section (woofer) is domi-
   nant, and above it the high-frequency section
   (tweeter) dominates.


2. Filter slopes: analog filters have characteristic stop-
   band, or rolloff, slopes, which are integer multiples
   of 6 dB/octave (or equivalently 10 dB/decade). The
   simplest type of filter is the first order, or
   6 dB/octave filter. In passive loudspeakers, the
   highest order filters in common use are third-order
   (18 dB/octave), whereas fourth-order
   (24 dB/octave) Linkwitz-Reilly filters are popular in
   active crossover implementations.

17.8.5.1 Effect on Maximum Output

The choice of filter slopes used in a crossover has a
number of implications for the performance of the loud-
speaker system. Generally speaking, crossover filter
characteristics will affect a loudspeaker’s maximum
output capacity, amplitude and phase response, and
directivity.
Since all transducers have a maximum excursion
beyond which their output is no longer linear (or perma-
nent damage occurs), and since the required excursion
for a given acoustic output level increases with
decreasing frequency, the characteristics of the
high-pass filter(s) in a crossover have a direct bearing
on a loudspeaker’s maximum available acoustic output:
in general, selecting a higher cutoff frequency will
reduce the excursion required of the high-frequency
transducer(s), as will employing steeper filter slopes.
For a given high-frequency transducer, increasing the
crossover frequency reduces the displacement required
of that transducer. The demand made of the woofer as a
result of the increase is strictly thermal, since the lower
end of its band of use is not affected by such a change.
This benefit has to be balanced against the possible
inability of the woofer to effectively radiate higher
frequencies over a large angle.
In addition to excursion limiting, the bandwidth of
the signal applied to a given transducer determines the
thermal load the transducer will see in operation. For
this reason, dividing the spectrum into a greater number
of bands—thereby reducing the total power that is
applied to any single band—can also increase the avail-
able acoustic output of a loudspeaker. One must
consider, however, that very seldom will the signal
applied to a loudspeaker contain a constant broadband
spectrum. At times, much of the energy applied to a
loudspeaker may be confined to a relatively narrow
range of frequencies. In such cases, the advantage of

having a greater number of loudspeaker bands is

substantially reduced.

17.8.5.2 Effect on Loudspeaker Response

The choice of filter slopes and alignments has major

implications for the response of a multiway loud-

speaker. Even though these effects have been examined

and published for decades, they are often either misun-

derstood or simply ignored by loudspeaker designers.

It is a good idea to state as simply as possible the

ideal functional requirements that should be met by a

crossover network: a crossover should enable the

acoustic sum of the individual transducers’ outputs to be

an accurate replica of the system input signal.

The response of a loudspeaker, for purposes of this

chapter, is defined as its pressure response at a partic-

ular point in space. Even though the above criterion is

simple to state, there are many design constraints that

lead to tradeoffs in a loudspeaker’s accuracy. For

example, prevention of damage to transducers is often

an overriding consideration in the design of a cross-

over. This may motivate the designer to consider steeper

filter slopes. In some loudspeaker configurations,

off-axis response anomalies are intrinsic to the design.

The designer may wish to make off-axis anomalies in

amplitude response as geometrically symmetrical and as

narrowband as possible. The Linkwitz-Reilly filter

family is sometimes employed in pursuit of these goals.

The simplest crossover is a first-order filter pair. In a

two-way loudspeaker, the first-order transfer functions

for low-pass and high-pass functions are:

(17-7)

and

(17-8)

where,

Fl is the low-pass transfer function,

Z 0 = 2Sf 0 is the angular cutoff (3dB) frequency,

S is the Laplace complex frequency variable,

Fh is the high-pass transfer function.

If we add the two electrical transfer functions, we get

(17-9)

F 1

Z 0

S+Z 0

=---------------

Fh S

S+Z 0

=---------------

Tt

S+Z 0

S+Z 0

=---------------

1=