Misinterpretation - cascading does not eliminate phase errors, only minimizes by correcting for HP or LP, depending on direction of cascade, but cannot correct for both simultaneously.
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Due to comments in the previous post above, I checked the reference at this link :
Woofer crossover & offset
In the text accompanying the last graph about cascading sections, the author states:
"The cascade filter topology works for any number of channels and is independent of how close the crossover frequencies are to each other. The outputs add to a flat amplitude response. It is an allpass with non-linear phase."
This is not true, as the following graphs indicate :
The error is significantly minimized, but cannot be completely eliminated.
Woofer crossover & offset
In the text accompanying the last graph about cascading sections, the author states:
"The cascade filter topology works for any number of channels and is independent of how close the crossover frequencies are to each other. The outputs add to a flat amplitude response. It is an allpass with non-linear phase."
This is not true, as the following graphs indicate :
An externally hosted image should be here but it was not working when we last tested it.
An externally hosted image should be here but it was not working when we last tested it.
The error is significantly minimized, but cannot be completely eliminated.
Here are simulations as above, but parallel connection:
An externally hosted image should be here but it was not working when we last tested it.
An externally hosted image should be here but it was not working when we last tested it.
Of course
One of the problems of a L/R in some applications such as large drivers with wide spacing.Off axis fall off of the lower frequency driver will compound the issue and increase the roll off away from the vertical axis even further beyond the theoretical slope, making the sweet spot even narrower in practice with the L/R.
Not so in all cases. If you use small drivers and/or a rather low crossover frequency then this is the case.
However an actual 3dB peak above/below axis does not lead to an apparent wide vertical sweet spot because you still have a 3dB change in response, which at a crossover frequency of 3-4Khz causes a massive difference in perceived "presence" and imaging.
Too much presence (which a 3dB hump definitely is in that critical region) is just as bad in a different way as not enough, and will sound wildly different to the on axis response.
I would consider this to be a flawed design, so for that reason I probably wouldn't use a 3rd order crossover on any design where driver directivity, spacing, and crossover frequency would allow the 3dB peak to manifest.
A wide sweet spot to me means a response that (especially at the crossover frequency) stays as uniform as possible over a given angle, without undue peaks or dips.
As I mentioned before if you match the lower frequency driver's natural change in directivity to the driver spacing and crossover frequency you can still get a flat on axis response but effectively "flatten" the 3dB off axis peak, so that instead of a narrow pointy lobe with 3dB peak you get a wide "bulbous" lobe that is relatively uniform near ~0dB over as much as about 30 degrees. (From around -10 to +20 degrees)
In the opposite direction where the response would tend rapidly towards a notch both the depth of the notch and the rate at which it approaches the notch is also significantly reduced by the falling off axis response of the lower frequency driver. (You can't get a full notch if the response of one driver is significantly down compared to the other at the off axis angle where the phase shift would cause the notch)
In other words, you're using the 3dB off axis peak of a 3rd order filter to enhance the angular size of the lobe in a design with large widely spaced drivers.
This all only works if the drivers have their acoustic centres aligned of course, if they were misaligned to redirect the peak of the lobe towards the horizontal, the delicate balancing act doesn't work, resulting in a 3dB peak on axis with a more rapid roll off in either direction.
It does work
And if you were implementing it passively, in a 3 way, say, your suggestion would mean that the woofer was being supplied through two low pass filters in series - the midrange low pass and then it's own. Not very practical when it's hard enough to get series resistance of the woofer crossover low with a single filter.
In the cascaded high pass approach, woofer and midrange go through the same number of filter sections as a parallel system, the only difference is the first high pass section of the midrange bandpass would also feed the input of the tweeter high pass, relatively easy to do, with a minor amount of tweaking of values to account for loading.
I'm not convinced that cascading low pass sections would cancel out the interaction of phase shifts in the same way as cascading high pass sections, but I haven't sat down to try to analyse it in detail so I might be wrong. I'll leave that for the experts...
So I guess Linkwitz is wrong and you're right ?
All I see is a bunch of graphs produced by some un-named program with unknown inputs with the word "Cascade" in the description. Until I see the exact filter topology that software is purporting to simulate, I'd rather go with what Linkwitz has to say on the matter
If you design with driver directivity as part of the vertical response this way, the horizontal polar and thus power response will change. The BW3, prior to driver directivity and non-coincidence issues are added, provides constant power. Did you measure the horizontal polar and approximate power response with this sort of design change for vertical response? I suspect that this will introduce more pronounced dips in power response around the crossover area and maybe some unusual horizontal polar response changes. Of course on balance it may all be a wash.
Dave
I'm not convinced that cascading low pass sections would cancel out the interaction of phase shifts in the same way as cascading high pass sections, but I haven't sat down to try to analyse it in detail so I might be wrong. I'll leave that for the experts...
So I guess Linkwitz is wrong and you're right ?
All I see is a bunch of graphs produced by some un-named program with unknown inputs with the word "Cascade" in the description. Until I see the exact filter topology that software is purporting to simulate, I'd rather go with what Linkwitz has to say on the matter
If you don't believe the results, do the math and see for yourself.
If you don't believe the results, do the simulations and see for yourself.
If you don't believe the results, build the circuit, measure, and see for yourself.
I have done this all and stand by the results.
Anyway, looking at the diagram for cascading, COMMON SENSE should tell you there will still be anomolies, since the residual phase error of the LP will still be present in the summed response.
The topic of this post was active crossovers.
This is another reason why passive xovers are not a good choice.
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to DBMandrake: you were right! (and I was wrong)
OK, this is a sort of "mea culpa" EMail, targeted at DBMandrake. We were having a disagreement about the location of the +3dB peak and the null for the Butterworth 3rd order crossover. My model was showing something different than some, but similar to others... so it was difficult to really pin down. In order to know exactly what is happening you need to know whether the tweeter is above or below the mid/woofer and the relative polarity of the two drivers' electrical connections. These are not always (usually NOT) stated in any of the references that I was able to find, that also showed a polar plot or the off axis frequency responses.
So, I turned to John Kreskovsky, who has done very similar modeling to look at the power response of difference crossovers, to ask him to help me out . A big THANKS to John for his help! He told me that indeed his plots are for tweeter above woofer, both connected with the same polarity, which showed results that were different (opposite from) my own result. So I went hunting again through my calculations and formulas and found that I had essentially applied the phase calculation backwards, resulting in the opposite behavior with respect to polarity. This wasn't evident to me at first - it really only changes things for the odd order crossovers. Since I wasn't familiar with the polar response flipping with polarity and which was which, I didn't pick up on the fact that the behavior was reversed.
So, I have to give some kudos for DBMandrake's "back of the envelope" calculations, that he made back in post #18 regarding the relative phase of the drivers. You were absolutely correct! Nice job.
Also, something John mentioned during our correspondence might be of interest to everyone. Regarding the position of the +3dB lobe in the Butterworth odd order responses:
Well, this was all a little discouraging. I had developed the formulas in my spreadsheet bit by bit, while increasing the capabilities, and this somewhat haphazard probably led to the error. Yesterday I decided to start fresh using a good definition of the locations of the sources and observation point, and ended up redoing pretty much everything, using a different way to calculate the phase and amplitude at the observation point. I also changed the position at which I am calculating the "off-axis" observations. Previously I was essentially moving "straight up or down" from the "on axis" point to reach a particular off axis angle, however this also changes the distance to each driver as well as the distance from the reference point midway between the drivers on the baffle. Now I am moving up and down in an arc that is on a spherical surface with a fixed radius, equal to the "listening distance". Should I ever want to expand teh spreadsheet to calculate power response, it is already in the right form for that.
So, now that I am taking in to account the actual distance from each acoustic center to the observation point, I found something interesting: when the listening distance is, say 1m, and the drivers are vertically separated by say, 0.05m, the relative amplitude for the tweeter and woofer within their passband is slightly unequal off axis at the observation point, increasingly more so as the angle increases. This is because the relative distance between the drivers and the observation point is starting to become slightly different and, since power falls as 1/R, one driver will be "closer" and thus "louder". When you move farther out, this effect goes away, but it was a nice confirmation that things are working correctly. The other result is that as you move farther away, the amplitudes in the passbands also decreases with 1/R, so I added the option of normalizing the response to 1/R.
So, again, thanks DBMandrake for being persistent, and for putting the splinter in my mind that forced me to track down this issue. The new version is a nice improvement on my earlier version, and corrects the mistake that you pointed out. Luckily this doesn't drastically alter most of the other results. I will update the white paper when I get a chance with the new plots and will include diagrams of the model system for completeness.
-Charlie
OK, this is a sort of "mea culpa" EMail, targeted at DBMandrake. We were having a disagreement about the location of the +3dB peak and the null for the Butterworth 3rd order crossover. My model was showing something different than some, but similar to others... so it was difficult to really pin down. In order to know exactly what is happening you need to know whether the tweeter is above or below the mid/woofer and the relative polarity of the two drivers' electrical connections. These are not always (usually NOT) stated in any of the references that I was able to find, that also showed a polar plot or the off axis frequency responses.
So, I turned to John Kreskovsky, who has done very similar modeling to look at the power response of difference crossovers, to ask him to help me out . A big THANKS to John for his help! He told me that indeed his plots are for tweeter above woofer, both connected with the same polarity, which showed results that were different (opposite from) my own result. So I went hunting again through my calculations and formulas and found that I had essentially applied the phase calculation backwards, resulting in the opposite behavior with respect to polarity. This wasn't evident to me at first - it really only changes things for the odd order crossovers. Since I wasn't familiar with the polar response flipping with polarity and which was which, I didn't pick up on the fact that the behavior was reversed.
So, I have to give some kudos for DBMandrake's "back of the envelope" calculations, that he made back in post #18 regarding the relative phase of the drivers. You were absolutely correct! Nice job.
Also, something John mentioned during our correspondence might be of interest to everyone. Regarding the position of the +3dB lobe in the Butterworth odd order responses:
With normal polarity the [position of the] Butterworth [+3dB lobe] go[es] below, above, below, above as the order goes 1, 3, 5, 7…. And of course, depending on x-o frequency and separation there can be additional lobes and nulls.
This is a good one to remember, since the position of that lobe (and the null, on the opposite side of "on axis") is important.
Well, this was all a little discouraging. I had developed the formulas in my spreadsheet bit by bit, while increasing the capabilities, and this somewhat haphazard probably led to the error. Yesterday I decided to start fresh using a good definition of the locations of the sources and observation point, and ended up redoing pretty much everything, using a different way to calculate the phase and amplitude at the observation point. I also changed the position at which I am calculating the "off-axis" observations. Previously I was essentially moving "straight up or down" from the "on axis" point to reach a particular off axis angle, however this also changes the distance to each driver as well as the distance from the reference point midway between the drivers on the baffle. Now I am moving up and down in an arc that is on a spherical surface with a fixed radius, equal to the "listening distance". Should I ever want to expand teh spreadsheet to calculate power response, it is already in the right form for that.
So, now that I am taking in to account the actual distance from each acoustic center to the observation point, I found something interesting: when the listening distance is, say 1m, and the drivers are vertically separated by say, 0.05m, the relative amplitude for the tweeter and woofer within their passband is slightly unequal off axis at the observation point, increasingly more so as the angle increases. This is because the relative distance between the drivers and the observation point is starting to become slightly different and, since power falls as 1/R, one driver will be "closer" and thus "louder". When you move farther out, this effect goes away, but it was a nice confirmation that things are working correctly. The other result is that as you move farther away, the amplitudes in the passbands also decreases with 1/R, so I added the option of normalizing the response to 1/R.
So, again, thanks DBMandrake for being persistent, and for putting the splinter in my mind that forced me to track down this issue. The new version is a nice improvement on my earlier version, and corrects the mistake that you pointed out. Luckily this doesn't drastically alter most of the other results. I will update the white paper when I get a chance with the new plots and will include diagrams of the model system for completeness.
-Charlie
Same problem I had, in quite a bit of searching I couldn't really find any on-line articles that were convincing either for or against, due to either ambiguity in the test setup (driver positioning and polarity not explicitly stated) or a seeming lack of understanding or due care and attention of the author.
I'm sure there are definitive AES papers etc which cover it, but they don't make for a good reference in a forum discussion for those (including me) that don't have access to them.
I got the impression that, in the hobbyist arena at least, odd order crossovers are seen as the "red haired step child" that nobody much cares about or takes notice of, so apart from briefly mentioning them they don't get much analysis or serious attention, with everyone looking instead at L/R designs.
The only reason I picked up on the error was that I was experimenting with a 3rd order design some 8 years ago for a ribbon tweeter, and went through all the same motions of calculating on paper the lobing effects with different polarities. (I didn't have any software to simulate it) From that I specifically chose the positive phasing to produce an upwards tilted lobe, despite knowing total phase shift is double in that configuration. (I also preferred the sound of the positive phased configuration after months of listening tests)
When I spotted the error I thought I'd point it out, but then when we got debating which way was right you really had me scratching my head and doubting myself, it'd been years since I'd worked out the phasing and I started to wonder had I got it wrong all those years ago, had I estimated the acoustic centres of those two drivers so wrong that there could be an additional 180 degrees shift due to improper alignment ? (At 4Khz 1/2 a wavelength, eg 180 degrees is only 43mm, and estimating or measuring the acoustic centre of a dual cone full range driver accurately at 4Khz when the voice coil is 60-80mm behind the frame is not as easy as it seems, especially when the apparent centre shifts with frequency)
There's nothing like having to defend your own belief of something to make you think it through more carefully and even learn new things in the process. I've always enjoyed the challenge of "thought experiments" where you try to reason your way through something and come out the other end with the (hopefully) right answer.
Yes, that sounds correct.
Indeed. This applies to any crossover, even L/R, especially at close listening distances - there is some asymmetry introduced into the lobe due to the fact that the drivers are no longer balanced in amplitude, and even a distance differential of a few centimetres makes a notice difference. It's nice to know your spreadsheet now models that too.
Do the vertical angles of the notches of a L/R and the notch/lobe of a 3rd order change angle slightly with measurement distance as well ? They will stabilize on a certain angle as you approach infinity, but when the driver spacing starts to become a significant fraction of the listening distance, there should be some shift. (With some large designs there will be a noticeable shift at a normal listening distance)
One affect with listening distance that won't be modelled by your spreadsheet (since it doesn't take driver directivity into account, nor can it easily with most real drivers since they don't have a nice predictable change in directivity once they go above their piston region) is further shifts in lobe angle with closer listening distances due to steeper / more un-equal angles from listener to driver.
At infinity the angle from listener to both drivers is the same, but as you get closer the angle to each driver starts to diverge dramatically especially where the height of the listener is between the two drivers, where the angle can be opposite.
Although it's easy to work out the angle from each driver to the microphone, I don't know how you would do anything useful with that piece of information unless you had a full directivity vs frequency profile of a driver. (I guess some speaker design sim software may incorporate this in their polar plots if they have enough data on the driver)
Great work though
Well, I'm coming at this from the opposite perspective - I'm not choosing a larger more directional low frequency driver specifically so I can try to "optimize" the off axis response of the 3rd order filter.
The choice of the two drivers was the starting point, chosen on other grounds, and their relatively wide spacing comes of necessity due to their physical size.
The full range driver's frame is 230mm in diameter, the ribbon tweeter 110mm, so even with the frames touching you're looking at a minimum centre to centre spacing of 170mm, add another 20mm or so to allow for the fact that to align acoustic centres the tweeter will need to be in a separate box atop the main cabinet, and you're looking at 190mm which is ~2.2 wavelengths at 4Khz.
So right off the bat this is not going to be a small driver closely spaced design with very wide dispersion at all frequencies, and it doesn't attempt to be, it's intended to be part of a large 3 way system with a listening distance of ~3 metres. Obviously the large driver will be somewhat directional at 4Khz, although a lot less than might be expected due to the whizzer cone.
Measured 30 degree off-axis response is quite impressive for an 8" driver, 4Khz fall off off-axis is fairly gradual out to 30 degrees, reaching just under 2dB down on the on axis response, then beyond 30 degrees it starts to fall off quite rapidly, in fact the whole ~2-5Khz range almost has something of a constant directivity characteristic.
Given that starting point the question becomes how can I optimize the directivity and sweet spot of those drivers at that wide spacing as much as possible with the crossover design.
So yes, there is naturally some loss in power response at and below the crossover point due to the driver itself, but the question is whether the crossover makes this worse than the driver by itself, or better.
Whereas the L/R will introduce a 3dB dip in the power response over and above the natural loss in power response of the drivers, the 3rd order butterworth will not add any additional power loss over and above the drivers themselves, provided the phasing is correct.
What I've found with these drivers with the 3rd order connected in phase, is that at no point is the 30 degree horizontal off axis response dipping below the natural off axis response of the full range driver measured by itself unfiltered, so there is a net improvement in the off axis response right through the crossover region.
There's no off axis notch centred around or near the crossover frequency. What I see instead is a very gradual fall in off axis around 2Khz due to the driver, then it gradually slopes up again towards 5Khz, with almost no discontinuity at 4Khz. Certainly there is no peak or notch near the crossover frequency, just a smooth upwards slope.
If I connect the drivers out of phase a small dip does form just below 4Khz. All these I measured years ago although not nearly as accurately as I would have liked to, unfortunately I no longer have those graphs.
My theory on what's happening is that if you connect the drivers in phase, this makes the low frequency driver the (90 degree phase) leading driver, so any additional phase lag (relative to the tweeter) up to a maximum of 90 degrees caused by increased high frequency roll off moves the relative phase of the drivers to being closer in phase as the amplitude is dropping, and gives a small counteracting "boost". Loss in off axis response is less than the one driver by itself.
If they're connected out of phase, the low frequency driver is the lagging driver to begin with, so extra phase shift makes it lag even further behind, so you have less amplitude and a relative phase shift tending from 90 towards 180 degrees, so the response is less than the one driver by itself.
I could be wrong, but that's how I rationalise what's happening.
I did try to take a few measurements yesterday with ARTA but had to give up and admit my current living room is simply too small and cluttered to take meaningful measurements of such subtle effects.
(I don't have access to an outdoor area either)Because the drivers are large and widely spaced this effect can't be measured accurately up close (minimum distance of at least ~1.5m I'd say) but the room is too small to do even a windowed measurement at that distance without severe room effects. When I do get a better place to take measurements one day I intend to take a lot more careful measurements and complete polar patterns and directivity profiles but until then it will just have to remain an interesting theory that I can't back up with solid reliable measurements...
The description of the diagram in the reference is definitly wrong. I encountered this same problem several years ago when I built a 3-way crossover using 4th order Linkwitz-Riley filters. When I discovered the summed signal had errors, I wasted several weeks trying to find "my problem". A friend of a friend teaches electronics and had a SPICE program so we modeled the circuit and it predicted almost exactly the errors I measured. We tried cascading both ways and both predicted peaks or dips depending on the driver's relative polarity. To be sure the filters were tuned right, we checked them separately as a 2-way, and the summed response was exactly what it should be.
There does not appear to be a way to contact Lintwitz Lab about this problem.
Couldn't find an email address on the linkwitz lab site, but did find sl@linkwitzlab.com listed here:
Linkwitz Lab - Company Information
I`ve been through a lot of this frustration during my 4-way project that ended up as a 1. order overkill low R parallel system. My experience is that there`s no other way to make the ultimate passive filter than just this by Mr. Linkwitz "wrong" way.
This means no parallel components,(exept from conjugate circuits) nothing that ruins pulse,phase or impedanse, just one serial component for each driver. The speakers even outperforms huge active systems (including the ones by Mr. Linkwitz) by it`s free, open airyness & dynamics.
I had to stop looking blindly at the nearfield meashurements done at SEAS and make up enough guts to focus on in-room meashurements & listening-tests.
The Linkwitz implemenation of cascading and not paralleling is not wrong, it does eliminate phase errors of the HP or LP filters, depending on whether HP or LP are cascaded. But it DOES NOT eliminate both simultaneously, as the diagram description incorrectly states.
1st order filters have the least amout of residual phase, so your implementation should provide least amount of summed response amplitude errors. Whatever filter order is chosen, the xover freqs can be slightly staggered to provide acceptable response.
Is it not a good idea to start with an xover that is as accurate as possible before driver interactions, etc. are taken into account?
1st order filters have the least amout of residual phase, so your implementation should provide least amount of summed response amplitude errors. Whatever filter order is chosen, the xover freqs can be slightly staggered to provide acceptable response.
Is it not a good idea to start with an xover that is as accurate as possible before driver interactions, etc. are taken into account?
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I've read the Linkwitz reference and I don't see it as an insurmountable issue.
The lament seems to be that any pair of adjacent sections can be designed for perfect sum but when all sections are added the summation "goes bad". This is to be expected. We are generally lucky with independence from section to section since there is adequate spacing between one crossover and the next. When the number of "ways" increases then we have to expect some interaction between alternate sections. Since the impact of the adjacent section is primarily phase then at the least an itterative approach would allow you to get good overall summing via compensating for the combined phase errors. For example 1st and second sections (bass and mid bass) add perfectly but the bass gives extra phase shift on the far side of mid bass. The midrange has to be added to that pairing with regard to the real (combined) phase. When adding the midrange you will need to consider its phase response over the mid bass and bass section's phase. Since the midbass is narrow band then its lower crossover is as important as its upper.
Does anybody remember the thread last year of the German guy that wanted to make a perfect 10 way? It had all the same issues. LR is only a perfect solution when sections are considered in issolation. Practical solutions are out there but you can't ignore alternate sections when bandpasses get narrow. Proportionately higher order crossovers should give a solution also.
David S.
The lament seems to be that any pair of adjacent sections can be designed for perfect sum but when all sections are added the summation "goes bad". This is to be expected. We are generally lucky with independence from section to section since there is adequate spacing between one crossover and the next. When the number of "ways" increases then we have to expect some interaction between alternate sections. Since the impact of the adjacent section is primarily phase then at the least an itterative approach would allow you to get good overall summing via compensating for the combined phase errors. For example 1st and second sections (bass and mid bass) add perfectly but the bass gives extra phase shift on the far side of mid bass. The midrange has to be added to that pairing with regard to the real (combined) phase. When adding the midrange you will need to consider its phase response over the mid bass and bass section's phase. Since the midbass is narrow band then its lower crossover is as important as its upper.
Does anybody remember the thread last year of the German guy that wanted to make a perfect 10 way? It had all the same issues. LR is only a perfect solution when sections are considered in issolation. Practical solutions are out there but you can't ignore alternate sections when bandpasses get narrow. Proportionately higher order crossovers should give a solution also.
David S.
Exactly opposite is true; as order increases, the peaks/dips become more severe. The best solution is the slightly stagger the frequencies. There are graphs in previous posts to this thread that demonstrate this.
The issue with too many crossover studies is that they tend to simplify matters and come to conclusions that aren't generally universal. Too many show electrical (mathematical) filter summations with no real driver delay or none of the added rolloffs and phase shift of the typical driver. When one crossover order sums well and another doesn't then the author concludes that this is universally the case.
When designing real paractical crossovers you will find yourself frequently breaking the rules and getting good results. For example I am firmly in the Linkwitz Riley camp, believing that crossovers should have a combined response 6dB down at the corner and phase curves that lie over each other. Symetrical nulls and freedom from upward or downward peaks, the end result, are to be desired. Still, the papers will tell you that you need Butterworth squared, hence even ordered crossovers and in practice this is often not the case. Since the inter-driver delay is arbitrary you are just as likely to find that a pair of, say, third order sections with a soft (-6dB) corner might be just right for getting the phase curves of two sections to overlap and therefore for the sections to add. It can be Linkwitz Riley in nature without following the exact recipe.
Now Linkwitz knows this and his simulations at the top of the referenced page delve into a subwoofer crossover with varying delay times to find combinations that work.
Returning to his description of the cascaded circuit, it seems that the performance differences between the cascade and the non-cascade are fairly trivial and not the only apparent solution. Ripple for the cascade solution is not far off the previous (note that this is a pretty expanded scale and errors are under 1dB total!). For both cases it appears that a little bit of crossover shift would make things as good as you need, but being a hypothetical analysis it seems we must stick with fixed crossover points. Again, if you were building up a real system from scratch the usual itteration approach would get you as close as desired without even knowing that you were going down a less perfect theoretical path.
I think the same is true of his 4 way illustration. Any two sections will add fine, but the presence of a narrow bandpass on the high side of a section will create a little extra phase shift on the opposite side. This doesn't mean that you won't find a way to stitch in the next section, just that the classic LR filters won't work perfectly without a little fudging.
David S.
When designing real paractical crossovers you will find yourself frequently breaking the rules and getting good results. For example I am firmly in the Linkwitz Riley camp, believing that crossovers should have a combined response 6dB down at the corner and phase curves that lie over each other. Symetrical nulls and freedom from upward or downward peaks, the end result, are to be desired. Still, the papers will tell you that you need Butterworth squared, hence even ordered crossovers and in practice this is often not the case. Since the inter-driver delay is arbitrary you are just as likely to find that a pair of, say, third order sections with a soft (-6dB) corner might be just right for getting the phase curves of two sections to overlap and therefore for the sections to add. It can be Linkwitz Riley in nature without following the exact recipe.
Now Linkwitz knows this and his simulations at the top of the referenced page delve into a subwoofer crossover with varying delay times to find combinations that work.
Returning to his description of the cascaded circuit, it seems that the performance differences between the cascade and the non-cascade are fairly trivial and not the only apparent solution. Ripple for the cascade solution is not far off the previous (note that this is a pretty expanded scale and errors are under 1dB total!). For both cases it appears that a little bit of crossover shift would make things as good as you need, but being a hypothetical analysis it seems we must stick with fixed crossover points. Again, if you were building up a real system from scratch the usual itteration approach would get you as close as desired without even knowing that you were going down a less perfect theoretical path.
I think the same is true of his 4 way illustration. Any two sections will add fine, but the presence of a narrow bandpass on the high side of a section will create a little extra phase shift on the opposite side. This doesn't mean that you won't find a way to stitch in the next section, just that the classic LR filters won't work perfectly without a little fudging.
David S.
Couldn't agree more. It's all too easy to look at a crossover from a purely mathematical or electrical perspective without considering the intimate symbiosis with the drivers and thus treat it as a one dimensional (electrical domain only) problem resulting in a very limited set of "optimal" solutions.
With that type of thinking it's easy to have a pet crossover type / topology based on some perceived benefit in it's mathematical/electrical response and not even consider other filter types or modifications to the design due to "flaws" that may only be relevant in the clean sterile world of maths and electrical signals - "if all you have is a hammer, everything looks like a nail"
It's not just the additional phase shifts and variation in frequency response of the drivers, but also as you mention acoustic centre misalignment - and a large percentage of speakers are made without trying to align driver acoustic centres (on the basis that it's just too hard with some drivers without unacceptable baffle structures or diffraction) so something has to be done to the crossover to account for this, thus "spoiling" it's idealized response in the name of a better acoustic result, or even choosing a different order completely, like odd over even as you mention.
Before you can even begin designing a crossover that will sound good you need to be intimately familiar with the drivers you'll be using and their foibles, some drivers will sound better with some filter types and some with others, and it's hard to know what will work best with a certain combination of drivers and physical construction without a bit of trial and error. If you're open minded enough to "stray off the reservation" the results can sometimes be surprising.
As you point out there is also a whole continuum between the formalized cookie-cutter crossover types such as butterworth, linkwitz-reily etc. They are all good starting points when considering a design but the reality is there is infinite variability available between them, variations that can be exploited to optimize the acoustic response without adding any additional components, such as varying the overlap or LC ratios to tweak the response at the crossover frequency.
Crossover design is a very iterative process to get the best results on anything other than a trivial design, (more so than some with a strong filter theory background would like to admit I think) and although it certainly helps to have a decent grasp of the concepts that are involved, you don't need to be a filter design wizard who can go into all the maths behind the poles and zeros to optimize a crossover.
These days if you have access to rapid iterative tools - for example spice circuit simulators, you can perform extremely rapid "what if" testing, such as generating a family of curves based on sweeping individual component values, making arbitrary changes etc, and you can very quickly home in on the result you want, together with accurate measurement software to verify the performance of the actual filter, and acoustic response.
At the end of the day some of the minor shortcomings of different electrical filters are so dwarfed by the flaws of the drivers and other effects such as baffle diffraction that it's wasted energy to get too hung up on small theoretical "rule breaking" with filter design - whatever design approach gives the desired acoustic response is valid in my opinion.