**Crossover Design**

This is unavoidably a fairly technical section. The topics are:

- A review of design criteria for a crossover (immediately below)
- different crossover topologies
- Transient response
- Frequency response
- The problem of driver impedance
- Impedance compensation the with "Zobel" and other networks
- The choice of crossover components
- Multi-amping, and the option of using passive RC filters
- The effect that room reflections have for different crossovers.

Caveat: it is nearly impossible to design a good crossover without good test equipment to measure speaker parameters and crossover performance. The parameters provided by speaker manufacturers are not all that reliable. Good test equipment is not cheap, but it is a good use of your speaker-building budget.

**Crossover design criteria **

Desirable characteristics of a crossover network are:

- Rapid response cutoff outside the crossover frequency limits, so a driver doesn't receive frequencies it can't reproduce well.
- Uniform overall system frequency response, to avoid artificial coloration of the music.
- Good transient response for crisp reproduction of impulsive sounds.
- Minimum resistive losses and distortion.

For equal power levels, the cone excursion grows as frequency decreases (it
is inversely proportional to frequency squared above cone resonance). Undesired
low frequency content can push the speaker into the non-linear magnet/voice
coil region, leading to distortion. (See discussion
and data in System Design section). Frequencies above the design range can
cause cone breakup, again causing distortion. Despite the big difference in
low frequency content between a 1^{st} and 4^{th} order, it
turned out that a direct measurement of my system showed virtually identical
2^{nd} and 3^{rd} harmonic distortion. But I intend to revisit
this issue in the future.

Minimizing losses and distortion within the crossover is mainly a matter of using good (read expensive) components; e.g. heavy air-core inductors and polypropylene capacitors. Further discussion regarding this below.

The design must take the characteristics of the real speaker drivers into account to achieve good performance; a standard cookbook crossover design based on idealized drivers just doesn't work well when it is connected to real drivers. Much more on this below. This is just one of the reasons that it is always a very good idea to measure the actual crossover response, with the real drivers attached, to catch any odd interactions, and to fine-tune component values.

A 2-way design simply uses one high-pass and one low-pass filter. The most common crossover topology is the "parallel" design, where the two wires of the amp output are connected to the input terminals of the low-pass, and to the input terminals of the high pass. Circuits have a "dual," where voltage meshes are replaced by current nodes, impedances with admittances, and voltages with currents. An example is a pair of conventional filters, and their duals [5.3kb]. The conventional filters for the parallel network structure are shown on the left, and the dual filters on the right. The parallel-network filters and their duals will produce exactly the same response across the load resistor R if the dual network filters satisfy:

- inductor values equal the parallel-network capacitor values times R
^{2}. - capacitor values equal the parallel-network inductor values divided by R
^{2}. - filters are driven by a current source equal to the voltage driving the parallel network, divided by R.

For a 2-way crossover, the middle branch is simply
eliminated. For an n-way crossover, more branches are added. For a 3-way (or
n-way) crossover there are other possibilities such as a tree
topology [3.6kb]. A diagram of the 1st order
series and dual-series designs [3kb] shows that for a 3-way they differ
in one connection. For a 2-way, the designs are the same. It took me a while
to recognize it, but the "series" topology is the dual of the "tree" topology.
There are other possible topologies, but I am not aware that they have any advantages
over those already considered. A performance comparison of topologies is shown
below. The tree and parallel topologies use the same filters. A table
provides component values [21kb] for several common filter designs. The
resistance R is the load (i.e. the driver) resistance in Ohms. For an "8 Ohm"
speaker this value is typically closer to 6 Ohms than to 8 Ohms. The crossover
frequency is f_{c} in Hz, the inductance L is in Henries, and the capacitance
C in Farads. For orders less than 4, the N/A in this table means remove it from
the circuit if it is in parallel with R, or replace it with a wire if it is
in series with R. The values in the table work for 2-way designs for the parallel
and tree topologies. For the series and dual-series, use the conversion given
above. The values in the table are the same as those given by Dickason, except
for the Bessel design, which follows a design in the Handbook
for Sound Engineers. For the tree topologies, the same component values
work for both 2-way and 3-way crossovers. For the parallel topology, the interaction
between filters alters the performance. Dickason provides modified component
values for this case.

Personally I think the biggest difference between
filters is simply the order. With a few qualifications, the cutoff rate for
all of the filters is 6, 12, 18, and 24 dB per octave for orders 1, 2, 3, and
4 respectively. Second order filters introduce a 180^{0} phase shift
between the high and low frequency bands. To avoid cancellation at the crossover
frequency, 2nd order filters are normally hooked up with the pair of wires to
one speaker reversed, providing a compensating 180^{0} shift.

The choice among filters for me boils down to either the ideal transient response of the 1st order Butterworth, or the rapid cutoff of the 4th order Linkwitz-Riley. I recently learned of a design generated and posted on the web by John K that provides a rolloff between a first and second order, and preserves phase. I personally have not taken the time to understand the design in detail, but I would have to add this to my list of candidates. All of the examples shown in the remainder of this section are for a 3-way system with crossover frequencies of 300 Hz and 3 kHz. They assume ideal drivers with flat frequency response, both amplitude and phase. Perfect time (phase) alignment is also assumed. The initial cases are for a constant, purely resistive, input impedance. This is to isolate the effect of the crossover network itself. Of course in the final design, the frequency response of the drivers should also be taken into account. The results are based on a circuit analysis, so (with one exception noted below) all impedance interactions between filters are included, both with resistive loads or real driver loads. Since power amps are normally voltage sources, all filters are driven by a perfect voltage source, including the series designs.

Good transient response means that an impulsive sound arrives at your ears at a single instant in time, as one nice solid jolt of sound pressure. Poor transient response means the system smears the sound over a few tenths of a millisecond, or into several impulses arriving at slightly different times. An excellent test for transient response is how a system reproduces a square wave. A square wave represents an extreme example of an abrupt transition in sound pressure, and contains a broad spectrum of frequencies. A sound system must reproduce these frequencies with a flat amplitude response, and with a flat (or linear with frequency) phase response, or the wave will no longer be square. This type of sound does not occur in real music. However a system that can accurately reproduce a square wave can also accurately reproduce anything that real music can throw at it. It is an acid test. [Thanks to Bob Stanton for pointing out my description of an ideal speaker response to a square wave was incorrect. See discussion in the section on Thiele-Small analysis]. The crossover network response to one cycle of a 1kHz square wave for a 1st order Butterworth shows perfect behavior [46.1kb] (this and a few other examples below are the same as those used in the Crossover Demo Section). There are four curves in the graph. The signals sent to the tweeter, midrange, and woofer are shown as the blue, green, and red curves respectively. The heavy black curve is the total response. With the assumed ideal drivers, this is simply the coherent sum of the voltages across the terminals of the three drivers. Frequencies above 50 kHz have been eliminated, causing the ripple in the response. Other than that, it is perfectly square. The responses shown are for the tree topology. The 1st-order series and dual-series topologies also produce a perfect square wave. The 1st-order parallel topology performance with the same filters (not shown) is good, but it is not perfect. It is also not perfect with Dickason's component values. Each time division in the figure is 0.2 millisecond. The human ear just begins to hear two distinct clicks when they are separated by half this difference, as discussed elsewhere. Even though ears do respond to time variations smaller than 0.1 millisecond, features closer in time than 0.1 millisecond tend to be blurred together in our perception of sound. In any case, variations within the time frame shown in the graph are perceptible, in principle. The square wave response of a 4th order Linkwitz-Riley is not a pretty picture [47kb]. A 4th order Butterworth, or the 4th order Dickason design, create a square wave response (not shown) that looks very similar. The big question is: can you hear the difference between a good and a bad square wave response? My own experiments indicate that for time-alignment the answer is yes, but for the crossover phase effects shown in these curves, no. The low-frequency response of the system (below 100 Hz) has essentially no effect on the edges of a 1kHz square wave; low-frequencies mainly effect the flat top part of the wave. At subwoofer frequencies it is shown in the section on room acoustics that, even with a perfect speaker system, the response is chaotic. I believe that transient response for woofer frequencies is almost certainly irrelevant, as far as the effect of a crossover is concerned. The enclosure can cause significant ringing, which may be perceptible.

The 1st-order Butterworth amplitude [33.4kb], and phase [32.8] response, again show ideal behavior. Other than the slow cutoff, the responses are perfect. The amplitude response is flat for both coherent (voltage) addition, and incoherent (power) addition. This is due to a relative phase shift of 90^{0} between the hi-pass and lo-pass filters, which occurs for all odd order filters. In any given direction the addition is always coherent for any speaker system, so the only physical meaning of the incoherent addition is that it approximates the response averaged over many directions. Again the responses shown are for the tree topology. The series topology response is the same. The dual-series topology has perfect phase and coherent response, and a 0.73 dB dip in the incoherent response. The parallel topology has a 0.73 dB and 1.45 dB bump in the incoherent and coherent responses, respectively. For the "A" tree, and series topologies the tweeter response rolls off at 12 dB per octave below the lower cutoff frequency. All other responses, and all responses for other topologies, decay at the 6 dB per octave rate. For the 4th-order Linkwitz-Riley, the amplitude response [54kb] shows the rapid 24 dB per octave cutoff. In this example the interaction between filters is not included - it will be shortly - so the coherent addition is perfectly flat, and the incoherent addition has a 3-dB dip at the crossover frequency. The phase response [52kb] shows why the square wave response is not good. The 4th-order filters interact differently in different topologies. A comparison shows [32kb] that the tree topology is the least sensitive to impedance interactions between the filters (I have not tried the duals of the tree networks of order greater than one). The parallel topology uses the component values given by Dickason for his "Fourth-Order APC (A) design" on page 113. The other topologies use values from the table presented above for the Linkwitz-Riley 4th order, which is similar to the Dickason design. (It might be possible to smooth out the performance by fudging component values. I didn't try). Results shown are for coherent addition; the incoherent addition response is similar.

These crossover designs, like all others I have seen, work (almost) perfectly if each filter is loaded by a pure resistance R. A plot of real driver impedances [38kb] shows a floor around the DC resistance of about 6 Ohms, a spike (or at least a bump) at the resonant frequency of each driver, and a rise at higher frequencies. (These curves are actually computed, but they virtually overlay measured data). The input impedance of a 4th-order Linkwitz-Riley crossover network loaded with these drivers is also shown. Note that in places it gets as low as 2-3 Ohms. First-order series and dual-series filters will always have a perfect coherent frequency response, and perfect square wave response, as long as one assumes a perfect voltage source and perfect drivers. However the demands placed on the drivers are different. The response of the individual drivers when loaded with the real driver impedances {42kb] shows that the drivers are being driven well outside their design region. The tree topology is also shown. The coherent response for the tree design is not perfect with real driver loads, and the demands on the drivers are the most severe. This indicates that the 3-way tree design should not be used without impedance compensation. The two types of series designs pose less, and similar, stress on the drivers. It is interesting to compare the driver responses for the series loaded with real drivers and the tree with resistive loads [39kb]. The total responses are equal, and perfect, in both cases. Compared to the tree, the series tweeter response drops much more rapidly, but the midrange is stretched a bit. It appears that the 1st order series design could work without impedance compensation, for some drivers. The response of the 4th order crossovers, loaded with the real driver impedances, shows major degradations [35kb] to say the least. Compared to the parallel topology, the tree topology is a little better, and the dual-series topology a little worse. Again the parallel network uses Dickason's values, and the others the Linkwitz-Riley values, which is similar to the Dickason design. There is also an interaction between the crossover and the enclosure design, discussed in the section on Thiele-Small analysis. I would be curious to know how many people are aware of how badly a crossover performs in this situation.

One cure for this problem is to add an impedance compensation network in parallel with each driver [1.4kb]. The left-hand side of this network is commonly called a "Zobel." The right-hand side is a notch filter. Zobels are pretty easy to design and rather fault-tolerant. Dickason provides the formulas

Re and Le are the voice coil resistance and inductance, in Ohms and Henries respectively. The notch filter is more problematic. It is not quite as easy to design, not fault tolerant, and for lower frequencies requires large inductors. Dickason also provides formulas for these components

The "Q"s in these equations are the standard Thiele
parameters, and fs is the resonant frequency of the driver. It is important
to measure the resonant frequency of the driver __in its enclosure__. The
notch is tuned to the resonant frequency, which depends on the enclosure.

What gauge wire is in your crossover? If you know, you probably built it yourself, because it is rarely specified. It's amazing to me that people spend money on monster cable, and then hook it to a puny inductor. The inductors in the 3-way 1st order Butterworth above are .32 mH and 3.2 mH. With 14 AWG the small inductor has .16 Ohms DC resistance, and with 10 AWG the larger inductor .32 Ohms. There is over 60-feet of wire in the smaller, and 300-feet in the larger inductor! A good power amplifier will have a damping factor of 200 or more, meaning that the output impedance is .04 Ohms or less (tube amps somewhat more). If you use 10 feet of 16AWG zip cord for speaker wire it also has .04 Ohms resistance ahead of the inductor, and an equal amount in the return leg following the voice coil. The amp output resistance, speaker wire resistance, and inductor resistance, essentially add together - each has the same effect on performance as the other (except for the tweeter where there is no inductor in series). The voice coil resistance of around 6 Ohms also has a similar effect, differing only by being on the other side of the crossover. In the example above, neglecting the voice coil resistance, and the half of the speaker wire following the voice coil, the inductor represents 67% and 80% of the resistance for the midrange and woofer circuits respectively. And these are pretty hefty inductors. Using 10 gauge speaker wire only reduces the resistance by 12.5% and 7.5% respectively. This is a no-brainer. Spend your money on heavy inductors, and use zip cord for speaker wire. Even in a bi-amped configuration the effect of .04 Ohms of speaker wire resistance is totally negligible. See the discussion in the section on Thiele-Small analysis. An inductor coiled around an iron core can be made much smaller and lighter; unfortunately iron cores cause hysteresis distortion. Only air-core inductors should be used. A nice review of crossover capacitor characteristics is given in Speaker Builder; Polypropylene capacitors are the best choice. Bipolar electrolytic capacitors are cheaper, but introduce a significant amount of distortion. There are some fancy capacitors that cost a lot more, but they really don't work any better. Two construction details: (1) inductor coils should be placed with their axes at right angles, and separated to minimize coupling, and (2) it is a good idea to individually fuse each driver following the crossover and compensation networks. This is simple, cheap, and provides maximum protection for expensive drivers. I have found Solen to have a selection of good quality crossover components (not to imply that there aren't others that are equally good).

Virtually all impedance problems disappear if each driver is driven by it's own amp. Good amplifiers are very tolerant of variable impedance loads (unless they drop below 1-2 Ohms or so). The other big benefit is that now the crossover goes ahead of the power amp, so resistance in the crossover network has virtually no effect on performance. The effect of interaction between crossover and enclosure is also greatly improved. See Small-Thiele analysis. The input impedance of an amp is typically 20 kΩ or more, so the crossover designs given in the table above are no longer practical, because the inductance values are too large. The most common approach is to use an active crossover. Kits are available from John Pomann and Rod Elliot. A 4th order Linkwitz-Riley design I built from a kit is presented and analyzed in a separate section.

For a digital crossover the logical topology is to obtain the crossover input directly from a digital device, and to follow the crossover by multiple digital-to-analog converters, which then feed multiple analog stereo channels. So this architecture also has all the advantages noted above. ( I suppose it could be considered a type of active crossover).

There is one other alternative - passive RC filters. These are dirt cheap, and have zero distortion. Amplifiers are a benign load for a crossover. All components operate at low power levels. There are two negatives: (1) the filter must be tailored to match the input impedance of a specific amplifier; (2) the crossover output voltage is reduced. The first negative means you probably need to build a new crossover if you replace the amp. For the second, since the loss occurs ahead of the power amp, it is only necessary to crank up the preamp gain. This is not a major problem as long as the loss is not more than 6dB or so. This option is particularly attractive for a bi-amped system using 1st order crossovers. I implemented this design in the original version of my own system [6.2kb]. The filter circuits [2kb] are simple. The component values for the 1st order Butterworth are

R_{A1} and R_{A2} are the input impedances of the high and
low frequency power amps respectively, in Ohms, and fs is the crossover frequency
in Hz. The voltage loss factor is equal to 1/(1+α). For example, for α=1,
the loss factor is 0.5, meaning the filter output voltage is 1/2 of the pre-amp
output voltage. The values of α can be the same for the high-pass and low-pass
filters, or different to correct for any gain imbalances. The loss decreases
with decreasing alpha, but so does the crossover input impedance. Most preamps
require a minimum load impedance to perform to spec. The minimum crossover input
impedance is given by

Setting this equal to the minimum required by the preamp creates a lower limit
for the values of α. I haven't worked out the theory for higher order designs,
but Luc Henderieckx has posted
a spread sheet for designing a 2^{nd} order Low-pass. My gut feeling
is that an active crossover is probably the way to go in more complicated situations.
I am using the active 3-way 4^{th} order Linkwitz-Riley in my current
tri-amped system, and managed to cob together a 3-way 1^{st} order passive
design by trial-and-error.

It may be a surprise to some, but room reflections have a different effect for different crossovers. Here's the proof.

Room effects are huge at low frequencies, typically causing variations of more than 20 dB over frequency spans of 10 Hz or so. Unfortunately it is essentially impossible to get rid of them. At higher frequencies the variations are more like ±3 dB, but it is possible to completely gate out the reflections. This provides the capability of directly calculating the changes produced by room reflections.

I connected my midrange all by itself to my system and measured the response with and without room reflections. I then took the difference between the SPL dB and phase response with reflections minus the response without reflections. This directly yields the change produced by room reflections. The process was repeated for my tweeter. The results for amplitude [42kb] shows hat there is a definite bias towards the reflections increasing the amplitude (room gain), which is as expected.The effect on the tweeter is larger, because it radiates a much wider beam than the midrange.

The phase change [42kb] is also significant.The phase tends to move in opposite directions for the tweeter and midrange, but they were both connected with the same polarity.The fact that the polarity is correct is confirmed by the fact that with the Linkwitz-Riley crossover there is a smooth transition in both amplitude and phase though the crossover region. A reversed polarity would produce a null.

The dB and phase differences were converted to a complex value of relative pressure. The theoretical response of an ideal crossover can then be applied to these changes to see the effect on the combined performance of the tweeter and midrange. In the absence of room reflections this procedure would yield a perfect response. Therefore this result shows the response that would be obtained with perfect drivers connected to perfect crossovers, but messed up by the actual reflections in my music room. The two xos are a 1^{st} order Butterworth and a 4^{th} order Linkwitz-Riley. The crossover frequency is 3kHz.As claimed, the result [41kb] is significantly different for the two xos. The effect is somewhat greater for the Butterworth, which one would guess is due to the larger frequency overlap between the two drivers. Everything is magnified at the low crossover frequency of 300 Hz, but for whatever reason in my system the LR response was worse than the Butterworth in this case.