3^rd Order 18db/ Octave Active Crossover Network

While I must admit that there are already quite a few crossover circuits on the ESP site, there's always room for another.  In this case, it's a rather unusual 18dB/ octave state-variable circuit.  There aren't many 3^rd order state-variable filters, and I don't know of anyone who's published a crossover network using this topology.

An 18dB filter has some advantages over the more common 12dB and 24dB/ octave designs, which must use a Linkwitz-Riley alignment to ensure there's no peak at the crossover frequency.  This isn't needed with odd-order filters, but the 1st order filter has a rolloff of only 6dB/ octave and few speaker drivers (especially tweeters) can handle the very slow rolloff at anything over 10-20W system power.  The midrange (or mid-bass) will often have undesirable breakup modes above 3kHz, and these aren't attenuated well with only 6dB/octave rolloff.

The most common are 24dB/octave (4^th order) Linkwitz-Riley alignments (although 12dB/octave, 2^nd order are also used), and these are the most common with electronic crossovers in general.  Some people don't care for the very rapid rolloff of a 24dB/octave filter, and when used in isolation, there are audible artifacts.  These disappear when both drivers (for 2-way) are connected, but there are claims in some quarters that they somehow 'ruin' the sound.  Predictably, no proof has been offered that I'm aware of.

Unlike even-order crossovers, odd-order versions do not require a signal inversion (aka 180° phase shift) for one of the outputs (nor does a 24dB/octave Linkwitz-Riley alignment).  A 'conventional' 3^rd order crossover is shown in Project 123.  However, this requires different values for the filters which makes it a bit more difficult to build.

You may ask why one would use a state-variable filter when Sallen-Key filters appear to use fewer opamps and are very easy to build (including using the P09 PCB).  The most compelling reason is that you only need one resistor value and one capacitor value for the selected frequency, making component matching far easier.  In reality, the number of opamps is the same for Sallen-Key filters when configured using a buffer between the 1^st order (first stage) and the 2^nd order second stage.  Use without a buffer is possible but not recommended.

This is a good question, and the answer is either "because you can" or "to listen to it" (and probably both).  There are countless claims about the audibility of different opamps (which are mainly specious), but few people perform controlled tests to determine the audibility of different crossover topologies.  While (in general) measurements tell you a great deal, there may also subtle effects when electronics and loudspeakers are combined.  In general though, measurements are preferred when loudspeakers are involved.

There are certainly differences between 'classic' Butterworth vs. Linkwitz-Riley filters, and while a measurement will reveal any response variations, a simple frequency response measurement cannot tell you how it sounds.  This is one area of electronics (actually electromechanical) where there is no known measurement that will reveal what a speaker system sounds like.  There's also a lot of personal preference involved - people have different ideas for how they prefer their music to be presented.  Flat frequency response is almost always the goal, and that requires measurements.

Few people will argue that all speakers sound the same, because this is clearly not the case.  Even apparently very similar systems with the same drivers can sound different, depending on the crossover network, cabinet construction, phase alignment, etc., etc.  There's no reason to expect that two crossover networks using the same xover frequency will sound the same either.  They might with some drivers, but they may sound different with others.

Any listening tests must (as always) be blind - if you know what you're listening to, subconscious bias and/ or the 'experimenter expectancy effect' can lead to you hearing differences that don't exist.  No amount of so-called 'critical listening' can overcome subconscious bias, regardless of the alleged credentials of the listener.  All non-blind tests are pointless, and the 'results' can usually be dismissed without a further thought.  Of course, there are some systems that are so different that anyone can tell them apart, but I'm talking about subtleties that may be such that any differences are tiny.  Measurements are the easiest way to tell which arrangement is best (and that's what I did).

One thing that you'll find very quickly is that building a 24dB/ octave L-R and a state-variable filter at the same frequency is irksome (to put it mildly).  You can get them to be close (e.g. 2.88kHz for L-R vs. 2.84kHz for state-variable).  This frequency difference might be enough to cause an audible difference.  The only way to be sure is to build one of each and compare them.  For what it's worth, P09 requires 10nF and 3.92k (because I have a box of them) to obtain 2.87kHz, vs. 10nF and 5.6k for the state-variable (2.84kHz).  The difference of 30Hz is not expected to be significant.

It should be pretty obvious that I don't need another active crossover network, since I have one (or more) of each type described in the projects pages.  Not all are in use - that would be silly, and I don't have the space for a multiplicity of speaker systems.  Those I have already take up more than their fair share of available workshop space!  Fortunately, I do have a 2-way test speaker with the 140mm woofer and 25mm tweeter connected directly to terminals.  This was used as the test enclosure when comparing the two crossovers, with a stereo amp used to provide the power.

The 3^rd order state-variable filter is fairly uncommon.  There are a few versions on-line, but most are not optimised for audio crossover applications.  The disadvantage is that a 3^rd order state-variable filter requires three tuning resistors and three caps.  This means that making it variable is generally not an option unless you have a supply of 3-gang pots.

The first opamp determines the gain and Q of the filter (set by resistors), and it conveniently allows the input to be reconfigured as balanced, without having to add a separate stage.  Fortunately, this also means that all resistors (other than those used for tuning) can be the same value (using two in parallel in two places).

If a balanced source has significant output impedance, there's a small loss of gain.  For example, if the balanced output (the source) has an output impedance of 1k (500Ω on each signal lead), the output level is reduced by a bit under 0.5dB - assuming 10k input and 'Q' resistors as shown.  If used unbalanced, an input buffer is mandatory unless the source impedance is no more than 100Ω.

Figure 1 - 18dB/ Octave 2-Way State-Variable Crossover

Fig. 1 shows the basic form of the filter.  The input stage (U1A) sums the input signal and integrator outputs.  There are three integrators in series, and the outputs of the two inner integrators are bandpass.  These outputs are not useful, as they are not only too narrow, but are also asymmetrical.  There's no point discussing these in any detail because they aren't used as outputs, they simply set up the conditions needed to get the high and low pass outputs at the required slope of 18dB/ octave.

Both high and low pass filters are presented simultaneously, displaced by 90° from each other.  At the crossover frequency, the outputs are at -3dB referred to the pass-band level.  It's the phase displacement that allows the outputs to sum flat, a phenomenon that's common to all odd-order crossover filters (including passive).  A Linkwitz-Riley filter (always even-order) has both outputs at -6dB at the crossover frequency.

Because all tuning resistors and capacitors are the same value, it makes 'perfect' tuning fairly easy, but the caps need to be selected, because 5% (or 10%) is not good enough.  Using 1% resistors is fine - there will be small 'disturbances', but they should be well below audibility.  Consider that an electronic crossover is orders of magnitude more accurate than even 'identical' loudspeaker drivers.  I've shown 10nF and 5.6k resistors, giving a crossover frequency of 2.84kHz.  This is pretty close to ideal for a simple 2-way speaker system.

One thing is noteworthy.  In the P123 (2-way) version, you still need four opamps, but you also need six tuning caps, where the Fig. 1 circuit here only needs three.  You also save one resistor, but since two of them require parallel resistors to get 5k, you really need eleven in all (not counting the 100Ω output resistors for either circuit).  The biggest saving is the caps, and finding three matched caps will always be easier than finding six.

State-variable filters are commonly used with 12dB/ octave (see Project 148) and 24dB/ octave.  Odd-order (1^st and 3^rd) are far less common, and as far as I know the 1^st order version was first published on the ESP website.  It's been copied at least once that I've found.  3^rd order state variable designs are similarly uncommon.  The number of integrators determines the filter order, so one integrator means 1^st order, two means 2^nd order, etc.  I've not seen a 5^th order design (30dB/ octave), but it can certainly be done (although I'm unsure why anyone would want to).

Things get more complex when state-variable filters are combined to get more than a basic 2-way crossover.  The P09 Linkwitz-Riley solves this by connecting the networks in such a way as to minimise the effects of phase-shift within the filters (and the opamps), but the same trick can't be used when a state-variable filter is employed.  Well, you could (maybe), but it will make the circuit far more complex.

Using a state-variable filter for 3-way or more will cause a small rise at the upper crossover frequency.  This is due to phase shift, and while the peak is typically small (around 0.8dB) it's undesirable.  It's almost certainly less than the peaks and dips from typical drivers, and it can be mitigated by using a simple phase-shift network if you think that's necessary (it's included below).

Figure 2 - 18dB/ Octave 3-Way State-Variable Crossover

The arrangement shown in Fig. 2 requires a very small phase shift created by the 1st order high-pass filter following the state-variable circuit.  Because of the second filter, you'll get a rise of about 0.8dB at the upper crossover frequency with the values shown, but it may be better or worse if the frequencies are changed.  The extra bit of phase shift introduced by R_p and C_p reduces the peak to a gentle 'ripple' with an overall amplitude of less than 0.1dB.  The trimpot needs to be adjusted for minimum amplitude ripple across the high xover frequency, or you can use a 3k resistor if the crossover frequencies are as shown.

If the circuit is expanded to 4-way no additional changes are needed, because the phase shift only affects the output when there's a connection from high-pass to low-pass filter sections.  There will be an effect between the low-mid and the high-mid sections, and it will be similar to that for the mid to high section.  Again, a phase shift network will eliminate any problems.  C_p will need to be scaled for the frequency (reduce the frequency by 10, and increase the value of C_p by 10).

Figure 3 - Electrical Response Of 3-Way State-Variable Crossover With Default Values

The response is shown using the tuning resistors/ capacitors shown in Fig. 2.  The frequencies are nominally 280Hz and 2.8kHz.  At the crossover frequency, each output is 3dB down, but the ±45° phase shift means that they sum flat.  The (barely visible) ripple is +0.063dB and -0.036dB (less than 0.1dB overall) and it will not be audible.  Without the phase shift network, there's a peak of about 0.8dB at 2.6kHz, and this may or may not be audible depending on your speaker drivers.

The phase shift needed is small but important.  Based on my simulations, the time difference between high and midrange outputs is about 11μs too short at 2.84kHz.  The compensation network creates a leading phase shift for the 'high' output, forcing the two signals to be exactly 90° apart.  The time difference (td) for a 90° phase shift can be determined easily ...

td = 1 / f / 4     So ...
td = 1 / 2.84k / 4 = 88μs

Note that both the 2-way and 3-way will normally be fitted with level controls and buffers.  Depending on the source, input buffers may also be required.  The buffers will ideally have some gain so you can increase the level is required.  If you know the actual speaker sensitivity at the crossover frequency (or frequencies), the levels can be preset to match the drivers.  For example, a tweeter with 90dB/W/m will require exactly half the level needed for a mid-bass or woofer rated for 84dB/W/m.

As noted above, if the filter is operated with an unbalanced input that has a variable or unknown source impedance, an input buffer is mandatory.  The filter Q is altered if R1 (Fig. 1) is increased by an external impedance.  The input buffer is a simple unity gain, non-inverting opamp stage as shown next.  These can be omitted if the filters will be supplied from a balanced source.  The 200Ω resistors at the opamp inputs are to minimise the likelihood of RF interference detection which may produce 'silly' noises if it's allowed to get through.

Figure 4 - Input Buffers

I've shown output level controls and buffers here, although they can be (almost) the same as those described for Project 09.  You can arrange something different if you prefer.  For example, you may want (or need) balanced outputs or high-current line drivers if the interconnects are particularly long.  I've shown two, but naturally you need four for a stereo 2-way system, and six for stereo 3-way.

Figure 5 - Output Level Controls and Buffers

These are as simple as possible.  Depending on the power amps, you may need to add output capacitors, as a DC coupled system is highly undesirable.  The 1μF caps shown are fine for a following impedance of around 22k (-3dB at 7.2Hz).  The low frequency output may need a larger capacitance if the impedance is less than 22k.  For the HF output, 1μF is fine for a following impedance down to 2.2k.

A faulty opamp (for example) may cause the output to 'go DC', and if that's amplified and sent to your speakers, the loss of magic smoke will result as they die (generating many expletives as they are expensive components).  If you happen to believe that caps are somehow 'evil', I strongly recommend a blind test.  If selected wisely they cause zero audible degradation.  Burnt speakers will seriously diminish your listening experience!

One thing I always try to do (and mostly I succeed) is to avoid 'weird' component values.  They aren't always truly weird, but any resistor beyond the E24 series (24 values per decade) becomes hard to get, especially if you only need two or three.  Where possible, I use the E12 series, but always with 1% tolerance.  Capacitors are generally available only within the E12 series, and it's easier to use only those that are stocked by hobbyist suppliers.  Not all values are included by all suppliers, so (for example) you may be able to get every value except 8.2nF.  This is annoying, but it's also a fact of life.

Likewise, the circuit should be easy to follow, especially if it's being built on Veroboard or similar.  I use Veroboard for all my prototypes, and difficult circuits can be a real pain.  Despite these limitations, the circuit must perform as described, without any misbehaviour with sinewaves, squarewaves or music.  This design is a little tricky to build on Veroboard, but it can be done as seen in the photo below.  After the prototype was complete, I tested it against a P09 (24dB/ octave) Linkwitz-Riley crossover, listening for any differences.  My workshop is not really the place for critical listening, but I was able to discern a difference.  In this case, the 18dB crossover was the winner, both with a listening test and by measurement.

This idea was developed initially to show how you can adapt the state-variable architecture to achieve something different from the usual and more common circuits that abound (both on my site and on the Net).  Not that there's anything wrong with most (all those I publish are built and tested), but this one is rather nice from an engineering perspective.  Some people are convinced that 4^th order (24dB/ octave) filters are 'too fast', and third order filters are a good compromise.  Overall, I'm rather impressed with the results, and it's probable that a PCB will be made available for a 2-way stereo crossover.

It's designed to use any opamp that you like (hence no opamp types are shown in the schematics).  If you use sockets, you can test with cheap and cheerful opamps, and upgrade if you think that will make a difference (mostly it won't).  Suitable types include the RC/MC4558, TL072, NE5532, OPA2134 and LM4562.  If you have a favourite for any reason, it can probably be used without problems.  Be aware that the NE5532 will have higher DC offset than the others, so an output capacitor is not optional.

Remember that if you test different topologies (or opamps) the test must be blind.  If you know which network/ opamp is in circuit you will hear things that don't exist.  I set up a full comparison test, using the same opamps in both xover networks.  In the photo below, I added visible labels for input, outputs and power.  The bypass caps can be seen - one 100nF ceramic below each opamp, and two electros.

As expected, the outputs sum perfectly flat - electrically.  The article Using Phase Shift Networks To Achieve Time Delay For Time Alignment is important reading, as the relative offset of the drivers in my test box is about 50mm.  This means that the tweeter should be delayed by about 145μs to obtain the required delay.  With the 24dB xover I had to reverse the phase of the tweeter to compensate (at least in part) for the offset.  For reasons that aren't entirely clear, the 18dB xover didn't require polarity inversion, and the response across the xover region was as flat as one can expect from any pair of loudspeaker drivers.

It's important to understand that this depends on the speakers you use, and their associated delay based on the acoustic centres.  Unless you have a means of ensuring that the drivers have the same physical offset between acoustic centres (sloped or stepped baffle, tweeter waveguide), some phase alignment will most probably be required.  You are dealing with electromechanical transducers, and measurements are the only way to ensure that the response is accurate.

Figure 6 - Veroboard Prototype of Crossover

The prototype was built as seen, and it includes an input buffer but no output buffers or level controls.  The test speaker I used it with has a woofer and a tweeter that's 6dB more sensitive, so the high-pass output was attenuated by 6dB.  I set it up alongside a Linkwitz-Riley filter (24dB/ octave, Project 09) with a switch that allowed me to change from one to the other, without knowing which was which until I heard (or thought I heard) a difference.  The two were also measured without moving the speaker or microphone so a direct comparison could be made.  The results are shown below.

The P09 board (24dB/ octave) was populated with just two opamps, and it used the input buffer from the Veroboard prototype to drive the filters.  The caps for both filters were selected - a tedious process for the P09, because it uses ten of them (vs. three for the state-variable).  I had to include an inverting buffer for the 24dB crossover as described above, and demonstrated in Fig. 7 and Fig. 8.  I normally don't worry about matching caps because I know that the error will be far less than the normal response of the speaker(s).  However, since I was comparing the two xovers I felt it was important to ensure they were as close as reasonable (within 1%).

Figure 7 - 24dB Linkwitz-Riley Crossover Response (Tweeter In-Phase)

The result is not as flat as one would expect, mainly because the acoustic centres of the mid-bass and tweeter are offset (the cabinet has a flat baffle).  The dip at ~2.5kHz is quite pronounced, and is most definitely audible.  This is despite my workshop being sub-optimal for critical listening with speakers.  I tried several methods to determine the acoustic centres of the two drivers, but none was especially useful.  This will be covered in an update to the article Using Phase Shift Networks To Achieve Time Delay For Time Alignment when I get the chance.

Figure 8 - 24dB L-R Crossover Response (Tweeter Reverse Phase)

By reversing the tweeter's polarity (180° phase-shift), the result is better, but there's still a noticeable dip.  It's only just audible in my workshop, but would be apparent in a 'proper' listening test.  There are two ways to eliminate the dip - one is to add a time delay (using phase-shift networks) and the other is to step the baffle so the acoustic centres of the mid-bass and tweeter are in-line.  Neither will happen with my workshop test speakers, so its default setup will be with the 3^rd order network.  You can see why from the next graph.

Figure 9 - 18dB (Fig. 1) Crossover Response (Tweeter In-Phase)

The characteristics of the 3^rd order filter gives better (almost perfect) flatness across the xover frequency.  There seems to be a good case for this rolloff, but it will be different with different drive units.  You can't take anything for granted when designing a loudspeaker system, but it's only the mid to high crossover that requires careful alignment of acoustic centres because the wavelengths are comparable to physical distances.  One wavelength at 2.8kHz is only 122mm, so a half-wavelength is 61mm.

I also compared the two measurements with the twin of the test speaker, with its included passive crossover network.  It's never used at high levels so I could get away with a series 6dB/octave filter.  Interactions between the mid-bass and tweeter are fairly pronounced, but despite that it manages to sound alright - not great, but acceptable for what I use it for.  I haven't included the measurement for that enclosure because it's not relevant to the topic.

This is an interesting circuit in a number of ways.  It looks like there are a lot of opamps for a 3-way version, but four dual opamps for a stereo xover isn't really especially complex.  If you were to build the same filter using Sallen-Key filters you'd need three dual opamps, so there's only a 'saving' of one package, but the resistor and capacitor values are much less friendly.

Another filter topology that can be used to get a similar response is the 3^rd order multiple feedback topology.  However, this will demand at least E24 series resistors, and that may extend to E48 in some cases.  While only one opamp is needed, it's a difficult circuit to get right, and not one that I'd recommend.  The values needed are truly irksome, and the design equations are daunting (to put it mildly).  The MFB filter topology is great for bandpass filters, but IMO it's rarely useful for high and low pass types because it always requires odd component values.

All things in electronics involve compromise, and the idea is usually to get the circuit to perform as required and minimise complexity and ensure that you get good results with a minimum of different component values.  Having nice, friendly frequency tuning resistors and capacitors is a great advantage, as it minimises the opportunity for mistakes and makes component selection far easier.

No two different sets of mid-bass and tweeters will ever be identical, and it's imperative that any design includes measurements.  For my test speaker, time alignment is essential with a 24dB L-R crossover, but with the 18dB crossover it proved to be perfect without me having to do anything.  This was an unexpected benefit, and not something I planned

As with any electronic crossover, no Zobel networks are required across the speaker drivers, but of course a separate amplifier is required for each driver.  This is not a challenge if you build your own amps, as it's easy to include the required four (or six) amps into a single chassis.  A recommended amplifier is the Project 127 (using TDA7293 power amp ICs) - the PCBs are stereo, so you need either 2 or 3 boards.  Should you prefer a discrete design, then P3A is suggested.  This amp has been a best-seller for 20 years, and is as popular now as it ever was.

As for this project, I don't know how many people will be interested in a PCB.  I may get a small number made and make a decision as to whether it becomes a standard item based on reader response.  If nothing else, it's an interesting idea that's worthy of a place in the ESP line-up.

There is not much to see here, as it seems that no one has published anything similar.  3^rd order state-variable filters are uncommon anyway, but using them for a crossover network doesn't appear on any searches I performed.

3^rd Order MFB Filter Calculator
3^rd Order State Variable Filter - English translation

The second is not a reference (it was found after I'd written this article), but it does show how one can get tied in knots trying to comprehend maths that are simply not needed to design a filter.  It's in Japanese which is not helpful.  The link shows the translated version, and it's a great pile of maths that aren't needed for design.  The 3^rd order filter circuit is very different from what we normally expect.  It's configured as a low-pass filter only.  There is a crossover, but IMO it's completely useless, as the rolloff is only 12dB/octave (despite half and double value components and a complex circuit).  It's a 'constant voltage' design that passes a squarewave (which sounds useful, but is not).

Main Index Projects Index