70 Crossover Types
Figure 4.9: Frequency response of Solen split 1st-order 1 kHz crossover; both filter outputs plus the sum, with one output reversed (phase inverted).
In passive crossover design, phase-inverting one of the outputs is extremely simple; just swap over the two wires to the drive unit in question; everything is done in the box, and no-one is any the wiser. Active crossovers, however, will need to use some kind of phase-inverting stage.
Figure 4.9 demonstrates that with one of the phases reversed we still get a dip, but it is now only 1.2 dB deep; information on this crossover scheme is scanty, but that is presumably how it is supposed to work. It might be possible to reduce the deviation from perfect flatness by partial cancellation of the dip with a response irregularity in one of the drivers. It is however still difficult to get enthusiastic about a crossover with 6 dB/octave slopes.
First-Order Crossovers: 3-Way
It is difficult to make a 1st-order 3-way crossover because the slow 6 dB/octave slopes do not provide adequate separation into three bands across the audio spectrum. There is in any case little point because drive units capable of handling the wide frequency ranges inherent in such a crossover would probably be equally suitable for a 1st-order 2-way crossover setup.
Second-Order Crossovers
The use of a 2nd-order crossover promises relief from the lobing, tilting, and time-alignment criticality of 1st-order crossovers, because the filter slopes are now twice as steep at 12 dB/octave.Avery large
Crossover Types 71
number of passive crossovers are 2nd-order, because they are still relatively simple, and this simplicity is very welcome, as it reduces power losses and cuts the total cost of the large crossover components required. Neither of these factors applies to active crossovers; the extra power consumption and the extra cost of making a 4th-order crossover rather than a 2nd-order crossover are very small.
All 2nd-order filters have a 180° phase-shift between the two outputs, which causes a deep cancellation notch in the response at the crossover frequency when the HF and LF outputs are summed. Such a response is of no use whatever, and the standard cure is to invert the polarity of one of the outputs.
With a 2nd-order crossover using Butterworth filters, this gives not a flat response but a +3 dB hump at the crossover frequency. As we shall see, the size of the hump can be much reduced by using a frequency offset; in other words the highpass and lowpass filters are given different cutoff frequencies. An offset factor of 1.30 turns the +3 dB hump into symmetrical amplitude ripples of ±0.45 dB, which is the flattest response that can be achieved for the Butterworth crossover by this method.
Second-order crossovers have much less sensitivity to driver time-misalignments because of their 12 dB/octave slopes. Vance Dickason [2] has shown that for a 2nd-order Butterworth crossover, a 1 inch time-misalignment gives errors of only fractions of a dB, while a 2 inch misalignment gives maximal errors of 2 dB. The corresponding figures for a 1st-order crossover are 4 dB and 10 dB respectively.
The frequency-offset technique can also be used to reduce the effect of time-alignment errors on the amplitude/frequency response.
A 2nd-order crossover gives better, though by no means stunning, protection of the drive units against inappropriate frequencies, less excitement of unwanted behaviour outside their intended frequency range, and less modulation distortion. Since one output has to be inverted to get a usable amplitude response, the outputs are in phase instead of 180° phase-shifted, and so there should be no lobing error, i.e. tilt in the vertical coverage pattern.
Second-Order Butterworth Crossover
The 2nd-order Butterworth crossover is perhaps the best-known type, despite the fact that it is far from satisfactory. A classic bit of crossover misdesign that has been published in circuit ideas
columns and the like a thousand times is shown in Figure 4.10. You take two 2nd-order Butterworth filters with the same cutoff frequency, one highpass and one lowpass, and there you have your two outputs. As before, the summing device represents how the two outputs add linearly in the air in front of the loudspeaker.
As Figure 4.11 shows all too clearly, this does not work well; in fact “catastrophic” would be a more accurate description. Each filter gives a 90° phase-shift at the crossover frequency, one leading and one lagging. The signals being summed are therefore a total of 180° out of phase and cancel out completely. This causes the deep notch at the crossover frequency seen in Figure 4.11.
Since a 180° phase-shift is the root of the problem, that at least can be eliminated by the simple expedient of reversing the connections to one of the drivers, normally the high-frequency one of the pair. Figure 4.12 shows the result—the yawning gulf is transformed into a much less frightening +3 dB hump centred at the crossover frequency.
In a passive crossover this reversed connection can be hidden inside the speaker enclosure along with the crossover components, but in an active crossover the issue is more exposed. You will need to either
Figure 4.10: A doomed attempt to make a crossover using two 2nd-order Butterworth filters.
Crossover 2nd-order Butterworth 1kHz-Small Signal AC-13-Graph
norm
5.0
0.0
−5.0
−10.0
−15.0
−20.0
−25.0
−30.0
−35.0
−40.0
100.0 |
freq |
1.0k |
10.0k |
|
|
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Figure 4.11: The frequency response resulting from the in-phase summation of two 2nd-order Butterworth filters: a disconcerting crevasse in the
combined response. The dashed line is at −3 dB.
Crossover Types 73
Figure 4.12: Second-order Butterworth crossover, with the phase of one output reversed; the crevasse has become a more usable +3 dB hump. The dashed line is at −3 dB.
build a phase inversion into the active crossover, which again effectively hides the phase reversal from the user, or specify that one of the power amplifier-speaker cables be reversed.Alot of users are going to feel that there is something not right about such an instruction, and building the inversion into the crossover is strongly recommended.
Clearly our +3 dB hump is much better than an audio grand canyon, but accepting that much deviation from a flat response is clearly not a good foundation for a crossover design. That hump is going to be very audible. It might be cancelled out by an equalisation circuit with a corresponding dip in its response, but there is a simpler approach. Looking at Figure 4.12, it may well occur to you, as it has to many others, that something might be done by pulling apart the two filter cutoff frequencies so the response sags a bit in the middle, as it were. There is no rule in crossover design, be it active or passive, that requires the two halves of the filtering to have the same cutoff frequency.
Figure 4.13 shows the result of offsetting each of the filter cutoff frequencies by a factor of 1.30 times. This means that the highpass filter cutoff frequency is changed from 1.00 kHz to 1.30 kHz, while the lowpass cutoff becomes 1.00/1.30 = 0.769 kHz. Crossover now occurs at −6 dB. With the phase inversion, the offset factor of 1.30 turns the hump into symmetrical amplitude ripples of ±0.45 dB above and below the 0 dB line; this represents the minimum possible response deviation obtainable in this way. Now the amplitude response is looking a good deal more respectable, if not exactly mathematically perfect, and it is very questionable whether response ripples of this size could ever be audible. You may wonder if the frequency-offset process has rescued the response with outputs in-phase; the answer is that it is not much better; there is no longer a notch with theoretically
74 Crossover Types
Figure 4.13: Two 2nd-order Butterworth filters with 1.30 times frequency offset, in normal and reversed connection. With the phase of one output reversed, the +3 dB hump is smoothed out to a mere ± 0.45 dB ripple. The normal-phase connection still has a serious dip 9 dB deep. The dashed line is at −3 dB; the two filter cutoff (−3 dB) frequencies are now different.
infinite depth, but the there is a great big dip 9 dB deep at the bottom, and such a response is still of no use at all.
The frequency offsets required for maximal flatness with various types of crossover are summarised in
Table 4.1 at the end of this chapter.
The power response for the 2nd-order Butterworth crossover with no frequency offset is shown in
Figure 4.14; it is a perfect straight line. This qualifies it as a constant-power crossover (CPC); it is a pity that the power response is of much less importance than the pressure response. It is important to realise that the power response is the same whether or not one of the outputs is phase reversed. When the two outputs are squared and added to get the total power, the negative sign of the reversed output disappears in the squaring process.
When looking at this power response plot, it is important to appreciate that each crossover output is still shown at −3 dB at the crossover frequency, meaning the power output from it is halved. The two lots of half-power sum to unity, in other words 0 dB.
Earlier we saw that a frequency offset of 1.30 times was required to get near-flat amplitude response. How is that going to affect the perfectly flat power response of Figure 4.14? The answer, predictably, is that any change is going to be for the worse, and Figure 4.15 shows that there is now an 8 dB dip in the power response.
Figure 4.14: Power response of 2nd-order Butterworth crossover with no frequency offset—the sum is perfectly flat at 0 dB. Each crossover output is still at −3 dB at the crossover frequency, meaning the power is halved. The two half-powers sum to 0 dB.
Figure 4.15: Power amplitude response of 2nd-order Butterworth crossover with 1.30x frequency offset—rather less than perfect, showing an−8 dB dip at the crossover frequency.
76 Crossover Types
As mentioned earlier, 2nd order filters have a 180 degree phase-shift between the two outputs. This is shown in Figure 4.16, where the phase of the sum lies exactly on top of the trace for the lowpass output. The phase of the sum is that of a 1st-order allpass filter, the same phase characteristic as that of the 1st-order crossover. This is inaudible with normal music signals. For reasons of space, phase and group delay plots are from here on only given for the more interesting crossover types.
The summed group delay for the inverted connection is shown in Figure 4.17. It has a level section at 226 usec and a gentle peak of 275 usec just below the crossover frequency. Note that the level section shows less group delay than the 1st-order crossover.
When we were looking at 1st-order crossovers, you will recall that it was said that no other crossover could solve the waveform reconstruction problem, i.e. to get out the waveform that we put in after summing the outputs. How does a 2nd-order Butterworth cope with the square wave reconstruction problem?
The answer from Figure 4.18 is that it fails completely, and inverting one of the filter outputs makes things even worse. The phase-shifts introduced by the 2nd-order filters make it impossible for reconstruction to occur. While it is not very obvious from Figure 4.18, the first and second cycles of the simulation are not quite identical; since we have a circuit with energy-storage elements (capacitors) and we are starting from scratch, it takes a little time for things to settle down so you obtain the
result for continuous operation. In some cases it is not uncommon for 20 cycles to be required to reach equilibrium. Not every writer on the subject of audio has appreciated this fact, and major embarrassment has resulted.
Figure 4.16: Phase response of 2nd-order Butterworth crossover with one output inverted; the outputs are always 180° out of phase. The phase of the sum lies exactly on top of that of the lowpass output. Highpass output shown before inversion.
Figure 4.17: The group delay response of a 2nd-order Butterworth crossover, one output inverted.
Figure 4.18: Attempted reconstruction of a square wave by a 2nd-order Butterworth crossover without offset. Total failure!