Handbook for Sound Engineers

624 Chapter 17

The two transfer functions add up to a constant,
independent of frequency. This is a desirable result,
since the outputs of the radiators in a multiway loud-
speaker are ultimately recombined by (acoustic) addi-
tion. The transfer function of our electrical sum implies
that, in a two-way loudspeaker with ideal, perfectly
coincident transducers and a first-order crossover, the
system transfer function would not depend on
frequency. We could, with some additional effort,
engage in the same exercise with higher-order transfer
functions. If we did so, we would find that, of all
symmetrical (identical low-pass and high-pass slope
and alignment class) filters, only the first-order pair
does not introduce phase or amplitude error or both to
the loudspeaker’s transfer function. The interested
reader will find detailed mathematical analyses of the
various crossover topologies in the references cited at
the end of this chapter.
One way of examining the effects of crossover filters
on loudspeaker response is to use circuit simulations to
model various aspects of the system’s behavior. This
method has the advantage of presenting a simple
graphic representation of the values being modeled
without requiring extensive mathematical skills for
comprehension.

17.8.5.3 Two-Way Crossovers

For simplicity, we will examine several aspects of
crossover performance in two-way systems. Then we
will point out some of the elements that must be altered
when three- or four-way systems are contemplated.
The chart in Fig. 17-46 is the impulse response of a
first-order crossover, including the input signal,
low-pass, high-pass, and summed signals. For
simplicity, the crossover frequency has been set at
1 kHz. The choice of crossover frequency causes no loss
of generality.
Note that, although low-pass filter has obvious delay
and the high-pass filter overshoots the input signal’s
return to zero, these effects perfectly cancel each other,
rendering the input and the summed signals identical.
This characteristic is unique to a family of crossovers
identified by Richard Small as “constant voltage cross-
overs.” The first-order filter set is the only symmetric
low-pass/high-pass filter pair that falls into this class.
By contrast, the summed second-order impulse
response shown in Fig. 17-47 contains significant devi-
ations from ideal.
Note that, in the summed signal (output) there is
overshoot on the return to zero, followed by a delayed
reaction due to the low-pass filter’s delay characteris-

tics. Viewed in the frequency domain, the second-order

summed low-pass and high-pass response has a perfect

null—i.e., a notch that is infinitely deep on a decibel

scale—at the crossover frequency. Its phase response

goes through a wrap of 360° centered at the crossover

frequency.

Fig. 17-48 shows the impulse response family of a

fourth-order Linkwitz-Reilly filter pair. It should be

evident from this series of graphs that the impulse

response of a loudspeaker may be compromised by the

designer’s choice of crossover filter topologies. Viewed

in the frequency domain, the Linkwitz-Reilly filter pair

exhibits ideal amplitude response (i.e., perfectly flat)

through the crossover range and elsewhere, but its phase

Figure 17-46. Impulse response family of first-order cross-

over.

Figure 17-47. Impulse response family of second-order

crossover.

(^0) 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10

10
Input
Low pass
High pass
Sum
0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
10

10
Input
Low pass
High pass
Sum