Yesterday I said that I would post on Monday a little more detailed look at driver offsets and how they affect crossover design. I referred to this as Asymmetrical Crossover Mysticism because it os rare that a group of coincidences combine to our advantage and simplify something that is otherwise very complex. My theory is that this proves that God loves DIY speaker builders so he arranged for this confluence of coincidences to work out on our behalf. (I’m kidding).

First of all let’s get acquainted with a textbook 4th order Linkwitz-Riley crossover. In the textbook example both drivers are coincident, or share the same acoustic center. Both highpass and lowpass sections are made of cascading two second order Butterworth filters to result in a fourth order roll-off. Both sections are 6dB down at Fc, or the crossover point, and are perfectly in-phase with each other resulting in a correlated summation of +6dB and flat summed response. Beyond this, the acoustic phase response of the lowpass and highpass sections are identical and follow the exact same line. This is called “Phase tracking”. In an ideal Linkwitz-Riley crossover the drivers are not just in phase at the crossover point but are actually in phase with each other over the entire frequency spectrum. The summed phase response of a 4th order crossover will rotate a full 360 degrees over the audio spectrum and be 180 degrees from zero at the crossover point.

Here is a graph showing the amplitude and phase response of a textbook 4th order Linkwitz-Riley crossover at 2kHz:

If speakers had no acoustic offsets between them then we would simply target a final acoustic response for both the lowpass and highpass sections that matched the textbook 4th order L-R crossover and everything would sum flat. Unfortunately, that is not the case, and almost without exception the acoustic center of the woofer lies some distance behind that of the typical dome tweeter that it is paired with. A very typical amount of offset is around one inch, or 25mm. Here is what happens though if place the acoustic center of the woofer 25mm behind that of the tweeter in our textbook LR4 example. You will note that the summed response now has a dip just above the crossover point and the higher you go in frequency the greater the difference there is in phase between the lowpass and highpass sections:

Now here’s where a confluence of factors roll together on our behalf. First of all we know that there is 360 degrees of phase rotation in the fourth order crossover. For each crossover order, beginning with a first order network, there is 90 degrees of phase rotation per order. This is easily shown mathematically based on the poles of the complex response. This one factor is not coincidental, it is mathematical. It is just so happens to work out very nicely for us when the other factors are considered.

What are they other factors? Well it is just a coincidence that due to the diameter, and therefore the off-axis response, of most midwoofers they work best when crossed-over around 2- 2.5 kHz. And, it is likewise a coincidence that we tend to use 1” dome tweeters whose frequency response and output capabilities require a crossover point in the 2-2.5 kHz region as well. It is also coincidental that more often than not these tweeters are beginning to roll-off as second order sealed systems at this same frequency. And finally, it is also a coincidence that the typical offset between these midwoofers and tweeters is in the 1” – 1.25” range. There is nothing that said that any of this had to be this way. It just works out that it is most of the time, and we simply take it for granted. ;)

What we are really taking for granted lies in the offset itself and the crossover point. As pointed out, nearly all of our two-ways crossover at around 2 – 2.5khz due to driver size and frequency response constraints. With a typical offset of 25mm the woofer’s acoustic center lies at almost ¼ of a period of the wavelength of the crossover frequency behind the tweeter. This means the woofer’s delay is the equivalent of almost 90 degrees of phase shift relative to the tweeter’s. So, in order to bring these two drivers back in to phase alignment at the crossover we need to adjust the roll-offs to accommodate this 90 degree phase shift due to the offset. Since we already know that a one order change in crossover slope results in a 90 degree phase shift at the crossover we use this to our advantage. The most means for doing this is to reduce the woofer’s acoustic roll-off relative to the tweeters by one order to restore the proper phase alignment between the drivers. This also helps to keep the crossover simple because we already pointed out that the tweeter is beginning to roll off as a second order system in this region, so all we need to be achieve a 4th order L-R acoustic response is a properly tuned second order electrical filter combined with the tweeter’s existing acoustic response. Doing this means that the woofer’s acoustic roll-off needs to be only third order, and that when the delay from the third order roll-off is combined with the delay from the offset the result will be that the woofer’s acoustic phase will be back in alignment with the tweeter’s phase, and despite the offset the system will behave as a 4th order L-R system. Here is a graph showing a modified 3rd order slope on the woofer, along with a 25mm offset, combined with a tweeter possessing a 4th order Linkwitz-Riley response at 2kHz:

Here we see that the summed response is essentially flat with both sections 6dB down at the crossover point, in phase with each other, and combining for the flat summation. We also see that from about 3khz down there is very good “Phase Tracking” between the woofer and tweeter. Above 3khz the phase continues to diverge due to the offset distance become large relative to the short wavelengths at these higher frequencies.

As I said, this is most common way to deal with offset issues because it takes advantage of several of the coincidences we discussed. The crossover remains very simple due to fact that the offset works out to a nice ratio of the crossover frequency’s wavelength and allows us to reduce the lowpass crossover order. The highpass section is likewise simplified due to tweeter’s natural roll-off. And the crossover point has been determined by the drivers themselves and the fairly narrow region of overlap that they give us to work with. It just happens to all work out very nicely on our behalf, proving that God blesses speaker builders.

Now, in an earlier exchange Jay posted that the same summation could be met by relaxing the tweeter’s roll-off instead of the woofer’s. Although that may seem counterintuitive since the woofer is the one with the extra delay, he is correct. Here is a graph showing the same example only this time it is the tweeter’s slope that has been reduced to 3rd order and the woofer’s remains as a textbook 4th order Linkwitz-Riley response at 2kHz.

You will see in this example that the summation is still a flat response, and that both drivers are 6dB down at the crossover point indicating that they are in phase at the crossover. There is, however, a difference between this method and the previous one, and that difference is in the phase response. In the second the slope change results in a shift in the relative phase between the drivers at the crossover point that brings them into a correct in-phase relationship. This phase relationship is very narrow though and only exists in a narrow window right around the crossover point. On either side of the crossover point the phase of both drivers diverge quickly away from each other, so there really isn’t any phase tracking at all using this method, and I referred to it as “Phase crossing” because the phase response of the two drivers crosses at the crossover point and then moves apart on either side of it.

Jay, however, states that the phase relationship at the crossover point is good enough and still results in the flat in-phase summation, and from a practical perspective, he is correct. I agree that this method will likely produce a flat summation and generalized 4th order L-R behavior around the crossover. So, yes it appears that you can go asymmetrical in your slopes by one order in either direction and still result in a flat summed in-phase crossover. The overall phase relationship is different outside the crossover region, but in reality this really won’t matter enough to matter. In the end I would have to agree with Jay. I do not know if it is more difficult to optimize the summation this way or not since the phase relationship is different, but it looks like it can work. And again, it works because of the nice confluence of coincidences that work together on our behalf to make a 90 degree shift one way or the other bring the drivers back to the correct phase alignment.

Maybe this should have been a blog entry? :rolleyes:

Jeff B.

First of all let’s get acquainted with a textbook 4th order Linkwitz-Riley crossover. In the textbook example both drivers are coincident, or share the same acoustic center. Both highpass and lowpass sections are made of cascading two second order Butterworth filters to result in a fourth order roll-off. Both sections are 6dB down at Fc, or the crossover point, and are perfectly in-phase with each other resulting in a correlated summation of +6dB and flat summed response. Beyond this, the acoustic phase response of the lowpass and highpass sections are identical and follow the exact same line. This is called “Phase tracking”. In an ideal Linkwitz-Riley crossover the drivers are not just in phase at the crossover point but are actually in phase with each other over the entire frequency spectrum. The summed phase response of a 4th order crossover will rotate a full 360 degrees over the audio spectrum and be 180 degrees from zero at the crossover point.

Here is a graph showing the amplitude and phase response of a textbook 4th order Linkwitz-Riley crossover at 2kHz:

If speakers had no acoustic offsets between them then we would simply target a final acoustic response for both the lowpass and highpass sections that matched the textbook 4th order L-R crossover and everything would sum flat. Unfortunately, that is not the case, and almost without exception the acoustic center of the woofer lies some distance behind that of the typical dome tweeter that it is paired with. A very typical amount of offset is around one inch, or 25mm. Here is what happens though if place the acoustic center of the woofer 25mm behind that of the tweeter in our textbook LR4 example. You will note that the summed response now has a dip just above the crossover point and the higher you go in frequency the greater the difference there is in phase between the lowpass and highpass sections:

Now here’s where a confluence of factors roll together on our behalf. First of all we know that there is 360 degrees of phase rotation in the fourth order crossover. For each crossover order, beginning with a first order network, there is 90 degrees of phase rotation per order. This is easily shown mathematically based on the poles of the complex response. This one factor is not coincidental, it is mathematical. It is just so happens to work out very nicely for us when the other factors are considered.

What are they other factors? Well it is just a coincidence that due to the diameter, and therefore the off-axis response, of most midwoofers they work best when crossed-over around 2- 2.5 kHz. And, it is likewise a coincidence that we tend to use 1” dome tweeters whose frequency response and output capabilities require a crossover point in the 2-2.5 kHz region as well. It is also coincidental that more often than not these tweeters are beginning to roll-off as second order sealed systems at this same frequency. And finally, it is also a coincidence that the typical offset between these midwoofers and tweeters is in the 1” – 1.25” range. There is nothing that said that any of this had to be this way. It just works out that it is most of the time, and we simply take it for granted. ;)

What we are really taking for granted lies in the offset itself and the crossover point. As pointed out, nearly all of our two-ways crossover at around 2 – 2.5khz due to driver size and frequency response constraints. With a typical offset of 25mm the woofer’s acoustic center lies at almost ¼ of a period of the wavelength of the crossover frequency behind the tweeter. This means the woofer’s delay is the equivalent of almost 90 degrees of phase shift relative to the tweeter’s. So, in order to bring these two drivers back in to phase alignment at the crossover we need to adjust the roll-offs to accommodate this 90 degree phase shift due to the offset. Since we already know that a one order change in crossover slope results in a 90 degree phase shift at the crossover we use this to our advantage. The most means for doing this is to reduce the woofer’s acoustic roll-off relative to the tweeters by one order to restore the proper phase alignment between the drivers. This also helps to keep the crossover simple because we already pointed out that the tweeter is beginning to roll off as a second order system in this region, so all we need to be achieve a 4th order L-R acoustic response is a properly tuned second order electrical filter combined with the tweeter’s existing acoustic response. Doing this means that the woofer’s acoustic roll-off needs to be only third order, and that when the delay from the third order roll-off is combined with the delay from the offset the result will be that the woofer’s acoustic phase will be back in alignment with the tweeter’s phase, and despite the offset the system will behave as a 4th order L-R system. Here is a graph showing a modified 3rd order slope on the woofer, along with a 25mm offset, combined with a tweeter possessing a 4th order Linkwitz-Riley response at 2kHz:

Here we see that the summed response is essentially flat with both sections 6dB down at the crossover point, in phase with each other, and combining for the flat summation. We also see that from about 3khz down there is very good “Phase Tracking” between the woofer and tweeter. Above 3khz the phase continues to diverge due to the offset distance become large relative to the short wavelengths at these higher frequencies.

As I said, this is most common way to deal with offset issues because it takes advantage of several of the coincidences we discussed. The crossover remains very simple due to fact that the offset works out to a nice ratio of the crossover frequency’s wavelength and allows us to reduce the lowpass crossover order. The highpass section is likewise simplified due to tweeter’s natural roll-off. And the crossover point has been determined by the drivers themselves and the fairly narrow region of overlap that they give us to work with. It just happens to all work out very nicely on our behalf, proving that God blesses speaker builders.

Now, in an earlier exchange Jay posted that the same summation could be met by relaxing the tweeter’s roll-off instead of the woofer’s. Although that may seem counterintuitive since the woofer is the one with the extra delay, he is correct. Here is a graph showing the same example only this time it is the tweeter’s slope that has been reduced to 3rd order and the woofer’s remains as a textbook 4th order Linkwitz-Riley response at 2kHz.

You will see in this example that the summation is still a flat response, and that both drivers are 6dB down at the crossover point indicating that they are in phase at the crossover. There is, however, a difference between this method and the previous one, and that difference is in the phase response. In the second the slope change results in a shift in the relative phase between the drivers at the crossover point that brings them into a correct in-phase relationship. This phase relationship is very narrow though and only exists in a narrow window right around the crossover point. On either side of the crossover point the phase of both drivers diverge quickly away from each other, so there really isn’t any phase tracking at all using this method, and I referred to it as “Phase crossing” because the phase response of the two drivers crosses at the crossover point and then moves apart on either side of it.

Jay, however, states that the phase relationship at the crossover point is good enough and still results in the flat in-phase summation, and from a practical perspective, he is correct. I agree that this method will likely produce a flat summation and generalized 4th order L-R behavior around the crossover. So, yes it appears that you can go asymmetrical in your slopes by one order in either direction and still result in a flat summed in-phase crossover. The overall phase relationship is different outside the crossover region, but in reality this really won’t matter enough to matter. In the end I would have to agree with Jay. I do not know if it is more difficult to optimize the summation this way or not since the phase relationship is different, but it looks like it can work. And again, it works because of the nice confluence of coincidences that work together on our behalf to make a 90 degree shift one way or the other bring the drivers back to the correct phase alignment.

Maybe this should have been a blog entry? :rolleyes:

Jeff B.

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