Trying to Understand the Practical Use of Linkwitz-Riley Crossovers

**Dennis H** · 25 November 2007, 14:53 Sunday

LR2, LR4, whatever, are acoustical targets, not electrical targets. You do whatever you have to do on the electrical side to make the acoustical response work. Add in phase problems due to acoustic center offsets and it gets even more complicated. You know you have the phase right when reversing one of the drivers gives a nice deep null, even if it means deviating slightly deviating from ideal LR SPL response. After all, keeping the drivers in phase through the crossover region is what the LR crossover is all about.

This is why you bought LspCAD.

**Maximiliano** · 25 November 2007, 14:57 Sunday

Big no no, here. LR2 is NOT a circuit, but an acoustic slope. The calculators you use do not provide real LR2 slopes due to the natural rolloffs of the drivers you use. For corrent LR2 implementation you need to ALIGN your drivers' responses to LR2 target curves IN your CAD. In most cases, you end up using a first order electrical filter to obtain LR2 slopes. You also need to use a slanted or stepped baffle or a delay network for proper phase tracking, which is difficult to achieve on a flat baffle due to the LR2's large overlap between drivers.

Max

**JonW** · 25 November 2007, 15:44 Sunday

Thanks guys. :T I see. I knew I must be missing some key piece of info. So it’s really just an acoustic target. Hmmm… I thought my reading meant that it’s an acoustic target hit by specific electronic criteria. At any rate, the benefits of LR are flat frequency response at the crossover, good phase passing from one driver to the next, etc. Aren’t we already trying to do that with every crossover we ever design?

So do you only know when you’ve actually hit an LR2 (or LR4) when you get the null from reversing the tweeter polarity? Or are there other things you’re looking for.

I’m trying to understand the thought process here. At least, I’m confused enough to *think* that’s what I’m trying to understand.

Sorry for the base questions. I’m still trying to understand the basics. I appreciate the help, though.

**Dennis H** · 25 November 2007, 16:06 Sunday

You can, for example, use the optimizer in LspCAD to adjust components so you hit an LR4 lowpass on the woofer. Same for the tweeter highpass. Then you can start fine-tuning things to get the flattest sum between the drivers and the best phase match.

**Deward Hastings** · 25 November 2007, 16:48 Sunday

The difficulty you face in modeling “textbook” LR crossovers (and, as an aside, I’d suggest that LR4 is “better” than LR2 for a number of reasons) is that “textbook” crossovers assume an invariant (resistive) load, not an actual driver (or other crossover elements). There is, however, a circuit/device that completely resolves the problem and removes the effect of driver impedance changes from the crossover equation . . . it is commonly implemented as and called an “amplifier”. If such a device is placed between the “textbook” crossover and the driver the “textbook” crossover generally works just as expected. The only other significant requirements are that the drivers used should have well controlled (correctable to flat) response for perhaps an octave past the crossover frequency, and no significant abnormalities (cone breakup etc.) until the driving signal is at least 40-50 dB down (which will depend on crossover slope . . . about 2 octaves with a LR4 although severe breakups can often be addressed with out-of-passband notch filters), and that drivers be either physically or electrically time aligned.

There are circumstances where driver rolloff can be incorporated into the crossover design . . . the most common involves placing a mid/woofer in sealed box of a size which gives a 2nd order rolloff with a Q of 0.7 (a Butterworth alignment). Adding a “textbook” 2nd order of the same f3 (also Butterworth) then yields a LR4 highpass response (assuming the use of an impedance correcting device as mentioned above) which can be mated with a LR4 lowpass on the woofer/subwoofer. Where the resulting f3 is not convenient (or if the box is tuned to the desired f3 but the Q is not 0.7) then a Linkwitz Transform circuit can be used to bring the box/driver to a 2nd order Butterworth alignment at the desired frequency. The same technique can be used for tweeter highpass filters where the tweeter rolloff is controlled by its enclosure volume.

**augerpro** · 25 November 2007, 18:26 Sunday

Originally posted by JonW

At any rate, the benefits of LR are flat frequency response at the crossover, good phase passing from one driver to the next, etc. Aren’t we already trying to do that with every crossover we ever design?

We all seem to be designing for those properties because we are all using LR slopes. Model your current project using Butterworth 3rd order slopes. You won't get flat summing, and you'll never get that deep null when reversing a driver polarity.

**rc white** · 25 November 2007, 19:49 Sunday

As outlined in the original paper by Linkwitz the idea of the L-R crossover is to remove lobing error, this is the effect that when sweeping through the crossover region the vertical lobe wanders up and down unless the drivers are acoustically in line and are in phase.

It is possible for instance to optimize the on axis response and still get considerable lobe wandering, this is what most programs of the cheap, (and some not so cheap), or free download sort seem to do, and I suspect calculate a phase plot from the Hilbert transform using the assumption of a linear phase system, (loudspeakers being close to linear phase but not quite).
Showing a single phase plot is not good enough since it is the phase shift of one driver relative to the other that matters in lobe steering, and a single combined phase plot can look very smooth and yet still have considerable amounts of it.

There are many people who would have you believe that the wonders of computer simulation have rendered the art of speaker system design down to that of a simple paint by numbers cookbook exercise, its not.

**gmikol** · 25 November 2007, 21:03 Sunday

I'm surprised no one has mentioned this...

...but in a 2nd-order crossover (and 6th ???), the high side needs to have reversed polarity from the low side. In your schematic, the woofer and the tweeter have the same polarity. Can you post a plot with the tweeter reversed?

Anther problem is that it looks like the tweeter response has a dip around 2500Hz, very close to your crossover region. I don't see the dip in the manufacturer's data, though.

Lastly, the woofer has a nominal impedance of 8 ohms, but the tweeter has a nominal 4-ohm impedance. I'd also try re-calculating the 1800Hz XO for the tweeter assuming 4 ohms instead of 8.

--Greg

**Maximiliano** · 25 November 2007, 21:13 Sunday

Originally posted by rc white

As outlined in the original paper by Linkwitz the idea of the L-R crossover is to remove lobing error, this is the effect that when sweeping through the crossover region the vertical lobe wanders up and down unless the drivers are acoustically in line and are in phase.

It is possible for instance to optimize the on axis response and still get considerable lobe wandering, this is what most programs of the cheap, (and some not so cheap), or free download sort seem to do, and I suspect calculate a phase plot from the Hilbert transform using the assumption of a linear phase system, (loudspeakers being close to linear phase but not quite).
Showing a single phase plot is not good enough since it is the phase shift of one driver relative to the other that matters in lobe steering, and a single combined phase plot can look very smooth and yet still have considerable amounts of it.

There are many people who would have you believe that the wonders of computer simulation have rendered the art of speaker system design down to that of a simple paint by numbers cookbook exercise, its not.

LR crossovers are not to remove lobing errors, which exist in any 2 driver 2-way systems. As Dennis said above, LR crossovers are about having in-phase responses through the xo region and resulting symmetric lobing patterns around the 0 degree axis. And using HBT generated minimum phase of all drivers to design a crossover gives an accurate result as long as the drivers' responses are tailored properly.

Max

**---k---** · 25 November 2007, 21:14 Sunday

If you cross a driver well before its natural roll off, won't the roll-off predicted by the acoustical model be correct?

**jkrutke** · 25 November 2007, 21:22 Sunday

A good read on power response in relation to crossover type was written by John Kreskovsky (the original John K) at his web site.

musicanddesign.com - musicanddesign Resources and Information.

http://www.musicanddesign.com/Power.html

musicanddesign.com is your first and best source for all of the information you’re looking for. From general topics to more of what you would expect to find here, musicanddesign.com has it all. We hope you find what you are searching for!

Vertical polar and power response plots are there and it's a good primer to understand power response and crossover choice. It's a very pro-butterworth article however. I personally don't use 3rd order slopes often because there are some trade-offs not really emphasized in that article. I don't feel glued to LR4 design, but I end up with it often for many reasons. I prefer LR2 myself, but the drivers able to do that are very rare.

**rc white** · 25 November 2007, 23:13 Sunday

L-R crossovers

I have here in front of me Linkwitzs 1976 AES paper...

"Active crossover networks for non coincident drivers"

and in the abstract he specifically states that the aim of what he is presenting is to prevent the main vertical lobe shifting in direction and increasing in magnitude with frequency.

Furthermore he shows that a high/low pass filter pair has to have an all pass characteristic to achieve this, that is they have to have the same denominator and sum to zero..

(hp + lp)/d =1

Unless a crossover has this property there will be lobe steering.

Looking at the crossover components used and the published plots of the drivers, there is no way that combination of driver and filter characteristics can satisfy the above requirement, that is the on axis amplitude response must be flat and the phase shift of the two drivers identical.

**dlr** · 26 November 2007, 11:32 Monday

Odd-order BW does sum flat on-axis

Originally posted by augerpro

We all seem to be designing for those properties because we are all using LR slopes. Model your current project using Butterworth 3rd order slopes. You won't get flat summing, and you'll never get that deep null when reversing a driver polarity.

One correction is that all odd-order BW crossovers do sum flat on the design axis. They will have lobing off-axis that can be inverted by inverting the connection. As you point out, this inverted connection for acoustically aligned drivers won't have a null. The latter still maintains the on-axis flat response, but the group delay will change IIRC. It's the even-order BW that does not sum flat on-axis, they have a 3db peak.

**rc white** · 26 November 2007, 14:24 Monday

In the end it is not possible to get a second order filter by cascading two second order ones.
Moving coil transducers are inherently second order, and the only way of getting true second order slopes in any crossover is by cascading a driver transfer function along with a biquad transfer function and then a second order function.

If however the natural roll off s of the drivers are sufficiently far from the crossover then you can get what is an approximation of a second order characteristic but it cannot be an actual one unless the biquad is included.

In the system described by John W the drivers have sufficient overlap so that what is usually regarded as a close enough match to the ideal characteristic, using standard L-R filters can be achieved, looking at the diagram it is the fact that the crossover components are not correct for the driver impedances is the reason why the result is as it is.

With a 1.8kHz. crossover and acoustic alignment the above mentioned system should have around 20degrees of lobe steering as you sweep from one octave above to below the crossover frequency. You can measure this by measuring the vertical directivity the same distance bellow and above the tweeter, the difference in directivity in degrees is the amount of lobe steering, (note that as pointed out by Linkwitz in the article I mentioned acoustic offset also causes lobe steering).
In the context of odd order Butterworth filters, these filter pairs have the necessary all pass characteristic, but a constant 90degree phase shift relative to each other, this causes lobe steering.
If however you compensate for acoustic offset, and then move the tweeter back one quarter wave of the crossover frequency, the result is no lobe steering and a crossover that sums flat in both amplitude and power, (and a very large step in the baffle that causes lots of diffraction).

**Forte_II** · 26 November 2007, 15:51 Monday

From what I have read... If you dont time align the drivers either physicly or electricly the benifits of a L-R alignment disappear. It is all in the link above.

**Deward Hastings** · 27 November 2007, 02:59 Tuesday

It should be informative that the Linkwitz designed crossovers for both ORION and PLUTO contain time-alignment (delay) filters. And independent channel delay is on the “features” list of even the most inexpensive digital crossovers. The block diagram of the ORION crossover and the associated discussion of it and crossover design in general on the linkwitzlab site make almost a textbook on how crossovers should be done, whether implemented analog or digitally. It is the way virtually the entire pro-audio world has done it for years.

**Dennis H** · 27 November 2007, 14:22 Tuesday

I think you guys are making a bit too much of the Linkwitz reference and the theoretical requirements for a "perfect" LR crossover. The man himself knows full well how to overcome the limits of passive design using real drivers in real boxes (or OBs) as these snippets from his passive prototype page show. Perhaps these aren't real LR crossovers even though they are designed by the real SL?

Dipole protos

http://www.linkwitzlab.com/proto.htm

Open baffle loudspeaker prototypes with passive equalization and passive crossovers

The slopes of the midrange and tweeter filters are different. The steeper slope for the tweeter provides additional phase shift in the crossover region to delay the electrical signal and to bring the acoustic output in phase with the midrange. Since the tweeter is mounted forward of the midrange the added electrical delay gives better summation of the two acoustic outputs.

Reversal of the tweeter polarity and the ensuing depth of the notch indicate the accurate addition of midrange and tweeter output through the crossover region in normal mode.

**rc white** · 27 November 2007, 19:45 Tuesday

Note that the design referred to by Dennis H in the Linkwitz link is an mtm design that has no lobe steering because of its geometry, despite what crossover is used.

"Quasi L-R" crossovers have been described by both Thiele and Leach, these have even and odd order low/high pass filters, this allows the phase curves of the drivers to converge and meet.

**Dennis H** · 27 November 2007, 22:17 Tuesday

Note that the design referred to by Dennis H in the Linkwitz link is an mtm design that has no lobe steering because of its geometry, despite what crossover is used.

SL has both an MT and an MTM on that page. My main point was it's possible to make a good-sounding speaker even if some of the fine points of the theory get violated. The real world is seldom as tidy as AES papers might imply.

**ch83575** · 29 November 2007, 11:51 Thursday

Originally posted by rc white

It is possible for instance to optimize the on axis response and still get considerable lobe wandering, this is what most programs of the cheap, (and some not so cheap), or free download sort seem to do, and I suspect calculate a phase plot from the Hilbert transform using the assumption of a linear phase system, (loudspeakers being close to linear phase but not quite).
Showing a single phase plot is not good enough since it is the phase shift of one driver relative to the other that matters in lobe steering, and a single combined phase plot can look very smooth and yet still have considerable amounts of it.

First of all, I think this is one of the best threads I have poked my head into lately. It is amazing the cumulative knowledge base that is available here.

As for the quote above, I find some of the implications interesting. So, is it possible for the actual phase of a driver to deviate from the HBT of the FR? I have always wondered when Scan-Speak meant when they say that their SD motors reduce phase distortions... maybe this is related? If there is a deviation from theory would it be accounted for if we measured actual phase instead of calculating it? If so, is this method used by any of the common measurement packages (Prixis, soundeasy, ARTA, CLIO)?

I am very interested because in my own listening tests I have found that phase tracking is extremely audible, kind of the 800lb gorilla in the corner of speaker design.

**cjd** · 29 November 2007, 13:31 Thursday

Originally posted by ch83575

So, is it possible for the actual phase of a driver to deviate from the HBT of the FR?

I think there is a lot at play, and I KNOW I don't know everything. However, a couple things I think I know relating to this:

Distortion will show up in response, and HBT derives from this. So, it could be as simple as this. Less distortion, less variation from "perfect flat", less phase mess. But that probably also applies to power response, as radiation is NOT linear (and PR is one way we attempt to quantify this). Also, I'm not sure that phase will deviate from HBT so much as the frequency response is not a fixed thing so a single HBT does not cover all the cases.

Then too, Hilbert has trouble with multiple radiation points, so we're already having to massage that raw data when we attempt to model a minimum-phase system and work with it.

I saw this working up the Khanspire - I actually had a crossover from HBT massaged IB measurements + baffle diffraction/step modeling, and it resulted in flat on-axis with the real in-box measured data as well. However, a little additional work produced better results, better phase tracking, better off-axis - overall, better integration.

So simple on the surface, but dig in and it's nowhere near simple. And I'm only scratching the surface, and know it. So much to learn.

C

**Dennis H** · 29 November 2007, 14:49 Thursday

So, is it possible for the actual phase of a driver to deviate from the HBT of the FR?

The problem with Hilbert derived phase is it's only accurate at the acoustic center of the driver and there's no published spec for where that might be. If you measure the real phase, you don't need to worry about how far behind the baffle the AC might be; it's all included in the measurement.

**dlr** · 29 November 2007, 15:13 Thursday

Some things said are not clear as to intent...

or I'm misreading them. There are some points that are incorrect, however.

Originally posted by cjd

Distortion will show up in response, and HBT derives from this.

Whoa! There is no connection between HBT and distortion. HBT is simply a method to calculate the phase response from a minimum-phase frequency response and vice-versa. Nothing more. Distortion, if of the linear variety and
minimum-phase (as most in not all of it is in raw drivers), will simply be a deviation in the frequency response and hence in the measured and HBT calculated phase, but still valid. If of the non-linear variety, then the FR will have the same issue, but the response will not allow for fully accurate HBT, since the FR will change with signal level and content. That's where good drivers with good motors show their advantage. The non-linear distortion is non-minimum-phase, so it's contribution to the FR will be an issue. If too high, then the HBT will have significant errors. Most FR curves typically represent a low-signal level, hence low non-linear distortion contribution. The HBT phase will track measured phase closely if the model is created accurately.

Originally posted by cjd

So, it could be as simple as this. Less distortion, less variation from "perfect flat", less phase mess. But that probably also applies to power response, as radiation is NOT linear (and PR is one way we attempt to quantify this).

I'm not sure at all what you mean by this.

Originally posted by cjd

Also, I'm not sure that phase will deviate from HBT so much as the frequency response is not a fixed thing so a single HBT does not cover all the cases.

Can you clarify this comment? HBT is simply the calculated phase of a single measurement on a specific axis. Measure at any off-axis point and the HBT phase will match that measured phase. The phase off-axis simply varies according to the FR off-axis.

Originally posted by cjd

Then too, Hilbert has trouble with multiple radiation points, so we're already having to massage that raw data when we attempt to model a minimum-phase system and work with it.

Again, could you clarify what you mean? What is the "trouble" with the HBT?

Originally posted by cjd

I saw this working up the Khanspire - I actually had a crossover from HBT massaged IB measurements + baffle diffraction/step modeling, and it resulted in flat on-axis with the real in-box measured data as well. However, a little additional work produced better results, better phase tracking, better off-axis - overall, better integration.

I infer from this that you're saying that you did a good job of modeling (critical to accurate HBT-generated phase) and that your design changes produced better integration. I don't see the connection with the HBT vis-a-vis "troubles" with it. On the contrary, it sounds like it worked well for you. The integration is a designer issue, not an HBT issue.[/QUOTE]

**ch83575** · 29 November 2007, 15:13 Thursday

Originally posted by Dennis H

The problem with Hilbert derived phase is it's only accurate at the acoustic center of the driver and there's no published spec for where that might be. If you measure the real phase, you don't need to worry about how far behind the baffle the AC might be; it's all included in the measurement.

I think I at least partially understand the idea of excess phase and its implications on this, but cant the HBT transform be used on measurements that are not minimum phase? I think soundeasy (that is what I use) uses a HBT on just about all phase data, but the "manual" doesn't really make perfect sense to me, so I am not exactly sure how the math works for such things. So if it is using HBT on all calculations involving phase... and HBT is introducing some amount of error, where does that leave us. Am I worried about nothing here, or could this influence the sound of a design?

cjd, the distortion thing makes sense to me. It is included in the FR by our measurement techniques, but doesn't really influence the phase of the output signal (or does it?). Are there any other mismatches between the FR and HBT derived phase? Is there such thing as "phase distortion" or is that a physical impossibility?

-Chad

**cjd** · 29 November 2007, 16:10 Thursday

Dave: your comments and questions point to lack of clarity on my part, but otherwise agree with what I have in my mind as far as understanding.

So, to try and clarify, or really to just say "thanks for clarifying what I was thinking"

Originally posted by dlr

Whoa! There is no connection between HBT and distortion. HBT is simply a method to calculate the phase response from a minimum-phase frequency response and vice-versa. Nothing more. Distortion, if of the linear variety and
minimum-phase (as most in not all of it is in raw drivers), will simply be a deviation in the frequency response and hence in the measured and HBT calculated phase, but still valid.

This is what I was attempting to say. What distortion may exist will show up in the frequency response - as it does so, it will therefore be "included" in HBT. Distortion is still something I do not fully wrap my mind around - I think I grasp it at a macro level but don't address it well when discussing with others. For that matter, so is ALL of this I guess.

If of the non-linear variety, then the FR will have the same issue, but the response will not allow for fully accurate HBT, since the FR will change with signal level and content. That's where good drivers with good motors show their advantage.

This is what I was trying to say with my comments attempting to address Scan's phase comments. Along with the first part. That is to say, good motor, more consistent phase response on any measurement axis, any drive level, etc.

Can you clarify this comment? HBT is simply the calculated phase of a single measurement on a specific axis. Measure at any off-axis point and the HBT phase will match that measured phase. The phase off-axis simply varies according to the FR off-axis.

Here, you've again stated what I was attempting to say, but far more clearly. Which is also what I guess I refer to when I say "the trouble with HBT". Which is probably not fair to HBT, nor really accurate. It's simply something that has to be understood - the trouble is perhaps that this is NOT always understood.

I infer from this that you're saying that you did a good job of modeling (critical to accurate HBT-generated phase) and that your design changes produced better integration. I don't see the connection with the HBT vis-a-vis "troubles" with it. On the contrary, it sounds like it worked well for you. The integration is a designer issue, not an HBT issue.

Modeling lacks things like surround reflections, not-quite-perfectly flush-mounted drivers, whatever. As well as all the assumptions. Some of this is probably a slight mis-guess on acoustic centers. Maybe I'm drawing inaccurate conclusions. Probably.

C

**cjd** · 29 November 2007, 16:12 Thursday

Originally posted by ch83575

Are there any other mismatches between the FR and HBT derived phase? Is there such thing as "phase distortion" or is that a physical impossibility?

HBT is derived from FR - as such, a mismatch is an error in calculation or application of HBT.

Phase distortion, I think, may address a number of things: variance from a perfect flat perhaps, variation over different listening angles or drive levels (as Dave pointed out). It's a somewhat vague term as it's used.

C

**augerpro** · 29 November 2007, 16:17 Thursday

As I understand it non-linear HD is not measured by MLS type signals, so it is not included in the response.

I've always thought the HBT should basically match the measured phase? I just got a new soundcard and am moving to teh method where all drivers are measured at the same distance on axis. My understanding from skimming teh SE manual is that a distance is added to the AC entry until the HBT phase matches the measured phase. At least that's how I remember it, I haven't used this new method yet so I could be wrong...It does seem to me though that you must be very careful of the measurement setup and distances, or very small errors will show up as big phase errors, but this I think is our fault, no the program used or HBT.

**Dennis H** · 29 November 2007, 16:30 Thursday

Originally posted by ch83575

I think I at least partially understand the idea of excess phase and its implications on this, but cant the HBT transform be used on measurements that are not minimum phase? I think soundeasy (that is what I use) uses a HBT on just about all phase data, but the "manual" doesn't really make perfect sense to me, so I am not exactly sure how the math works for such things.

Minimum phase is what the HBT calculates from the frequency response. If your measurements only have two columns, frequency and dB, you need to run the HBT to calculate phase. If the measurement has 3 columns, freq, dB and degrees, the third column is the phase and you don't need to calculate it again.

Excess phase is the difference between the HBT calculated minimum phase and what you actually measured. Think of it as being due to the time of flight between the acoustic center and the mic. As the frequency goes up, a certain amount of time equals more wave cycles so you see the phase wrapping through 360 degrees faster and faster on the graph.

Different CAD programs deal with excess phase in different ways, e.g. subtracting a certain time of flight. I don't know how SoundEasy does things. The important thing to remember is the CAD packages can deal with excess phase when calculating filters. It's not necessary to subtract exactly the right amount of time (or any) from the measurements as long as both drivers are done the same way.

**dlr** · 29 November 2007, 16:30 Thursday

The best thing to do is...

Originally posted by augerpro

As I understand it non-linear HD is not measured by MLS type signals, so it is not included in the response.

I've always thought the HBT should basically match the measured phase? I just got a new soundcard and am moving to teh method where all drivers are measured at the same distance on axis. My understanding from skimming teh SE manual is that a distance is added to the AC entry until the HBT phase matches the measured phase. At least that's how I remember it, I haven't used this new method yet so I could be wrong...It does seem to me though that you must be very careful of the measurement setup and distances, or very small errors will show up as big phase errors, but this I think is our fault, no the program used or HBT.

follow John K's guide for SoundEasy.

**rc white** · 29 November 2007, 16:30 Thursday

In general optimizing for flat on axis response is easy, and optimizing for close phase tracking between drivers is not that hard.
If we require both to be optimized simultaneously, as we do to get an ideal L-R characteristic, then there is always a solution provided we incorporate a biquad transfer function.
If we don't want to incorporate a biquad then the only exact solution is when the driver roll off characteristics can form second order sections of an allpass hp/lp filter pair.
As pointed out by both Leach and Thiele there are approximate solutions that occur at particular combinations of driver and crossover functions where the phase curves converge, and you can swamp driver transfer functions to a certain extent if you incorporate an elliptical function such as in the "NTM" crossover.

**dlr** · 29 November 2007, 19:42 Thursday

It doesn't have to be considered in the extreme

Originally posted by rc white

In general optimizing for flat on axis response is easy, and optimizing for close phase tracking between drivers is not that hard.
If we require both to be optimized simultaneously, as we do to get an ideal L-R characteristic, then there is always a solution provided we incorporate a biquad transfer function.
If we don't want to incorporate a biquad then the only exact solution is when the driver roll off characteristics can form second order sections of an allpass hp/lp filter pair.
As pointed out by both Leach and Thiele there are approximate solutions that occur at particular combinations of driver and crossover functions where the phase curves converge, and you can swamp driver transfer functions to a certain extent if you incorporate an elliptical function such as in the "NTM" crossover.

Exact solutions are seldom necessary and not always a significant advantage. In fact, a time-delay for alignment coupled with a bi-quad will not provide as good an overall response to include power response if the drivers compared to drivers that are physically aligned with a good approximated FR, say to about 40db down, easily done. On the design axis, the time-delayed biquad has a theoretical advantage, but in reality the contribution in the deep stop-band is of diminishing audible influence.

However, physically aligned drivers with good standard crossovers, approximates if you care, will have a better integration off-axis, the major contributor to power response. This is because in the off-axis, the physically aligned drivers will show better phase tracking than the time-delayed bi-quad does since the time delay requirement changes in the off-axis for any drivers not physically aligned. This applies to both the horizontal and vertical off-axis directions. The non-aligned vertical lobing "wander" could, in fact, be more dramatic than for the aligned, standard setup.

For physically aligned, bi-quad setups you have the best theoretical start point.

Another consideration is the that this bi-quad setup may have some theoretical advantage, but the fact is that the advantage most often is at the point where the driver output is in the noise floor, so it gets into the area of being a moot point.

All of this is to the point of being a bit esoteric as the small differences in these comparisons may not be audible when comparing good implementations of both approaches.

**ch83575** · 30 November 2007, 13:18 Friday

So... kind of back to the original topic at hand. I think it is kind of pointless to argue wether it is better to physically align centers, or use delay methods or something else... since most of us are actually going to use a flat baffle anyways and the topic of the post is a "practical use" of the LR filter. So, if we start with LR targets, but deviate from symmetrical slopes in order to phase align the drivers on the flat baffle are we going to re-introduce the "lobe wandering" that the filter was designed to reduce? Or do the benefits of the filter remain intact due to the -6db crossover point or something else like that? Also, what exactly is this lobe wandering? Is the lobe at a different angle say at the woofer axis as opposed to the tweeter axis, or is it something that happens as you rotate around a single axis? Is the wandering itself upsetting aurally, or does it negatively effect power response? Guess I still have a lot to figure out, I wish I remembered enough calculus to actually read (and understand) the AES papers.

-Chad

**rc white** · 30 November 2007, 16:19 Friday

Lobe wandering or steering is an effect that when you sweep a tone from bellow to above the crossover frequency the vertical lobe starts to tilt as you approach the crossover frequency.

If the tone is such that the lower driver predominates then the lobe sticks straight out perpendicular to the baffle, as it does if the high driver predominates, when both drivers contribute the lobe tilts reaching a maximum tilt where both drivers contribute equally.

If the tweeter is above the woofer the lobe tilts downward, this can result in variation in floor reflection, and if for instance a singer is singing through a scale that encompasses the crossover, differences in timbre and location can result from variable reflection.

Manufacturers such as Mission for instance make speakers with the woofer on top so the lobe tilts up instead of down, reducing this effect.
Spendor make a three way monitor with the mid above the tweeter for the same reason.

I have heard that Linkwitz has conducted tests that show this effect is clearly audible, although I cant find reference to it on his site.

**JonW** · 30 November 2007, 18:48 Friday

I just want to say thanks to all for the excellent discussion. :T I don't quite follow all of it. So I need to take some time (later, when work lightens up!) to try and understand everything that is being said here. I learn so much from everyone at this board. :T

Trying to Understand the Practical Use of Linkwitz-Riley Crossovers

Trying to Understand the Practical Use of Linkwitz-Riley Crossovers

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