78  Crossover Types

The 2nd-order Butterworth crossover does not sum to flat even with one output inverted, though frequency offsetting can give a big improvement. The 12 dB/octave slopes are usually considered inadequate.

Second-Order Linkwitz-Riley Crossover

The Butterworth crossover filter can be made very nearly flat by tweaking the cutoff frequencies of the two filters.An alternative and much better approach in the 2nd-order case is to alter the Q’s of the filters. Setting the Q of each filter to 0.5, with identical cutoff frequencies, turns the 2nd-order Butterworth crossover into a 2nd-order Linkwitz-Riley crossover with each output −6 dB at the crossover point, as seen in Figure 4.19. The flat response qualifies it as an allpass crossover network.

The two outputs are still 180° out of phase for the normal connection and give the same yawning gulf in the response as the Butterworth. With the reversed connection the signals add as for the Butterworth case, but they are now 3 dB lower, so there is no hump. The summation gives a completely flat response, without the ripple you get with a frequency-offset 2nd-order Butterworth crossover.

The reversed connection has a −3 dB dip in the power response at the crossover frequency. In this respect it is not as good as the 1st-order crossover, which has a flat power response as well as a flat voltage (or SPL) response. However, as we have seen, the power response of a crossover is usually a minor consideration.

Figure 4.19: The frequency response of a 2nd-order Linkwitz-Riley crossover. The in-phase summation­ of two filters still has a cancellation crevasse, but with one output reversed the sum is exactly flat at 0 dB. The dashed line is at the −6 dB crossover level.

Crossover Types  79

The 2nd-order Linkwitz-Riley crossover is neither linear-phase nor minimum phase. The phase response plot looks indistinguishable from that of the 2nd-order Butterworth crossover, though there are minor differences. The summed group delay does not peak but rolls off slowly around the crossover frequency.

Second-Order Bessel Crossover

The Bessel filter has a much slower roll-off than the Butterworth but also has a maximally flat group delay; in other words it stays flat as long as possible before it rolls off, while the Butterworth group delay has a peak in it. This makes the Bessel filter an interesting possibility for crossovers with flat group delay characteristics.

The 2nd-order Bessel crossover without any frequency offset gives a −8 dB dip for the in-phase connection, as in Figure 4.20, but a more promising broad +2.7 dB hump with one output reversed, as in Figure 4.21. This looks very like the Butterworth hump in Figure 4.12, so it seems very likely we can also reduce this one by applying frequency offset to the filter cutoffs.

The first attempt; a frequency offset of 1.30 times, as used in the 2nd-order Butterworth case, reduces the size of the hump to +1.1 dB at its centre, but this time no dips below the 0 dB line have appeared; this differing behaviour looks ominous and suggests that the hump may get flatter and flatter with increasing offset but never actually reach a definite maximally flat condition.

Figure 4.20: The frequency response of a 2nd-order Bessel crossover summed in-phase has a dip going down to −8 dB. The dashed line is at −3 dB.

80  Crossover Types

Figure 4.21: The frequency response of a 2nd-order Bessel crossover summed with one output phase reversed has a +2.7 dB hump. The dashed line is at −3 dB.

However, this is a great example of a situation where you should not give up too soon. If we keep increasing the offset, then the shape of the summed response changes, until at an offset ratio of 1.45 we obtain a dip and two flanking peaks, as in Figure 4.22. The deviation from 0 dB is less than ±0.07 dB, so this result is actually much better than the 2nd-order Butterworth crossover, which at maximal flatness had deviations of ±0.45 dB. Such small deviations as ±0.07 dB will be utterly lost in drive unit tolerances. This offset ratio gives crossover at −6.0 dB.

The phase response plot looks very similar to that of the 2nd-order Butterworth crossover, but the rate of change around the crossover region is slightly slower due to the lower Q of the filters. The summed group delay does not peak but rolls off slightly more slowly than the Butterworth around the crossover frequency.

The 2nd-order Bessel is not linear-phase, though it deviates from it less than do the 2nd-order

Butterworth or Linkwitz-Riley types. It is not minimum phase. Avery good discussion of Bessel crossovers is given in [4].

Second-Order 1.0 dB-Chebyshev Crossover

All Chebyshev filters have ripples in their passband response, and given the problems we have had achieving a near-flat response when we were using filters without such ripples, things don’t look too hopeful. Using 1.0 dB-Chebyshev filters, which in 2nd-order form peak by 1 dB just before roll-off,

Figure 4.23: The frequency response of a 2nd-order 1.0 dB-Chebyshev crossover. In-phase summation gives a deep dip of −14 dB. The crossover is at 0 dB, because the cutoff frequency of this filter is defined as the point in the roll-off where the response returns to 0 dB after the 1.0 dB peak.

Figure 4.24: The frequency response of a 2nd-order 1.0 dB-Chebyshev crossover. Summation with one output phase reversed gives a peak of +6 dB. Vertical scale has been moved up by 5 dB to accommodate the peak.