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Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

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  • Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

    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.
    Click here for Jeff Bagby's Loudspeaker Design Software

  • #2
    Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

    Thanks for posting this Jeff.

    Greg

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    • #3
      Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

      An alternative method for dealing with the offset issue is to add delay to the tweeter signal to compensate for the offset. This is easily implemented with digital active crossovers or by adding an all-pass in an active analog design. Phase tracking is preserved. It is still necessary to address the phase shift caused by the tweeter lf roll off . . . that can be done by adjusting the delay (at the expense of phase tracking), or by incorporating the tweeter roll off in the crossover slope with a bi-quad, or in some instances simply by using a lower order (electronic) slope. Delay is also sometimes used for lobe steering . . .

      In passive designs where the crossover is used not just for signal splitting but also for driver response corrections (which always introduce phase shifts of their own) . . . well . . . it's still an "art" . . .
      "It suggests that there is something that is happening in the real system that is not quite captured in the models."

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      • #4
        Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

        Originally posted by Deward Hastings View Post
        An alternative method for dealing with the offset issue is to add delay to the tweeter signal to compensate for the offset. This is easily implemented with digital active crossovers or by adding an all-pass in an active analog design. Phase tracking is preserved. It is still necessary to address the phase shift caused by the tweeter lf roll off . . . that can be done by adjusting the delay (at the expense of phase tracking), or by incorporating the tweeter roll off in the crossover slope with a bi-quad, or in some instances simply by using a lower order (electronic) slope. Delay is also sometimes used for lobe steering . . .

        In passive designs where the crossover is used not just for signal splitting but also for driver response corrections (which always introduce phase shifts of their own) . . . well . . . it's still an "art" . . .
        Of course with active circuits you can add delay and compensate for the offset. This is true.

        Now remember, passive elements are all minimum phase. Any circutry used to flatten the frequency response will also flatten the phase response and move it in the direction of less phase error as well.
        Click here for Jeff Bagby's Loudspeaker Design Software

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        • #5
          Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

          Thanks Jeff, I copy and pasted into word for future reference.
          "I have not failed. I've just found 10,000 ways that won't work." Thomas A. Edison

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          • #6
            Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

            Originally posted by Jeff B. View Post
            Now remember, passive elements are all minimum phase. Any circutry used to flatten the frequency response will also flatten the phase response and move it in the direction of less phase error as well.
            That's not always the case in practice, though. Cone breakup, the most common inducement to apply notch filters, hardly shows up on a phase curve at all, but a notch filter used to suppress it most certainly will. Other examples are response correction to compensate for driver beaming, and baffle step correction (which is rarely applied globally as it should be, but is commonly included instead in the low-pass filter).
            "It suggests that there is something that is happening in the real system that is not quite captured in the models."

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            • #7
              Thanks Jeff!

              That is a (relatively) concise overview and much appreciated. I feel that many times people with strong opinions who are largely in agreement are done a disservice by the lack of communication inherent in this nuance free one directional electronic exchange we generously refer to as communication ;) . Many conflicts would resolve themselves in the presence of both parties, good music, and a respectful klink of a couple of beverages.
              When you run make sure you run,
              to something not away from, cause lies don't need an aeroplane to chase you anywhere.

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              • #8
                Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                Great write-up, Jeff. I say Blog-it, Blog-it, Blog-it ... and add your other articles if possible. The blog feature is pretty neat and I hope to see people using it.

                Seems that the additional difference between the two methods is that the target is no longer LR4 @ 2kHz. The first method maintains the target and the second does not. I would not consider that the end of the world, but when I design a crossover I am trying to hit a target.

                - Brad
                Brad
                piano black sealing mdf irregular recesses grill technique

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                • #9
                  Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                  Originally posted by Deward Hastings View Post
                  That's not always the case in practice, though. Cone breakup, the most common inducement to apply notch filters, hardly shows up on a phase curve at all, but a notch filter used to suppress it most certainly will. Other examples are response correction to compensate for driver beaming, and baffle step correction (which is rarely applied globally as it should be, but is commonly included instead in the low-pass filter).
                  Hi Deward,

                  It is always the case.

                  It must always be the case.

                  The reason natural (mechanical and passive electrical) systems have a minimum phase relationship with their magnitude function is that events passing through the system would show up before they actually occur (acausal).

                  For instance...

                  If you take system with a peak in it's transfer function and then overwrite it's phase response so that it's flat then it's impulse response will ring before and after the impulse.

                  Transfer function (frequency/phase response) is everything no matter what has effected it.

                  No separate considerations for tweeter rolloff, diffraction loss or anything else.

                  Comment


                  • #10
                    Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                    Originally posted by AJ View Post
                    Thanks Jeff, I copy and pasted into word for future reference.
                    Dually-noted!
                    Wolf
                    "Wolf, you shall now be known as "King of the Zip ties." -Pete00t
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                    • #11
                      Practical factors considered

                      Jeff showed, with excellent clarity, that a phase mismatch in LR4 2-way designs due to a driver offset on a flat baffle can be compensated by relaxing either the woofer or the tweeter's rolloff. Thanks, Jeff, for taking the time to write a good summary. What I'm going to add in the following is that in practice with real drivers and crossovers, we often end up using sort of an in-between approach of the above two methods. That is, it is very likely to "relax" both drivers' rolloffs to achieve flat summation or symmetric lobing in an LR4 design with a driver offset.

                      I'll take an actual design and simulate some different crossovers to show how a desinger can achieve a desired driver summation with practical factors considered. The example is Zaph's SR71. This design uses the Seas ER18RNX 7" and 27TDFC 1" drivers. If we use theoretical, textbook LR4 rolloffs with a crossover point of 1.75 kHz, we will have a response with a vertical polar response tilted downward due to the driver offset. On-axis summation does not have a full 6 dB gain at the crossover point as shown below:



                      In this simulation, I used a 30 mm driver offset for the 7" midwoofer. As expected, a phase mismatch occurred. Now, to compensate the mismatch, only the woofer's rolloff is relaxed:



                      As you can see, the phase mismatch has been compensated very well. But there are two practical problems with this approach:

                      1) Some people may want to suppress the cone breakup node at 4.3 kHz a little more.
                      2) The 4th order acoustic HP rolloff requires a 3rd order electrical filter. An additional capacitor may not matter, but if we really don't need it, then why have it? And some people want to design as simple a crossover as possible.

                      If we relax both drivers' rolloff---a little steeper rolloff on the woofer than the above and a slightly relaxed rolloff on the tweeter---, we obtain the following response:



                      As you can see, the phase alignment is excellent around the crossover frequency. The breakup node is attenuated a little more than the case of only the woofer's rolloff being relaxed. And this crossover requires only 2nd order electrical filters for both drivers.

                      Many existing designs' crossovers belong to this category. For example, Jeff's Dreydel 2-way uses a 3rd order electrical filter for the RS150 to give a sufficient attenuation to breakup nodes. As a result, the midwoofer's acoustic rolloff is not relaxed enough to fully compensate the phase mismatch. But he used a 2nd order filter on the tweeter, which results in a slightly relaxed tweeter rolloff. By doing this, he achieved good phase alignment at the intended listening axis.

                      In sum, to obtain a good LR4 phase alignment for drivers mounted on a flat baffle, we can relax either the woofer or the tweeter's rolloff rate. Relaxing only the woofer's rolloff gives a better compensation than relaxing only the tweeter's rolloff. But for practical reasons, we often end up relaxing the tweeter's rolloff slightly as well. The reasons are:

                      1) By doing this, we can use a bit steeper rolloff on the woofer's response and still have good phase alingment. This is useful to attenuate the woofer's breakup nodes.

                      2) Even a slightly less steep HP rolloff than a true 4th order slope can provide sufficient protection for today's well-designed tweeters.


                      In this sense, the "asymmetricity" may not be viewed as the key element of this method. It is "relaxation" that makes this method work. How about coining a new term "relaxed LR4?" ;)

                      -jAy

                      Notes

                      1. In my above sims, irregular ripples at 1.5 kHz to 4 kHz are mainly due to the diffraction effect on the tweeter's response. "Relaxation" of the rolloffs is not the major cause of them.

                      2. Although the relaxation occurs out of the crossover region, it has an effect upon the summation in the crossover region due to a phase shift caused by it. If you apply this "relaxing" technique to LR 4th order designs, its effect out of the narrow crossover region is not that significant, because individual drivers' SPL in regions where the actual relaxation occurs is already too low to affect the system SPL to a significant degree.

                      3. In theory, you can also apply this technique to LR type crossovers of higher orders. For example, you can relax the rolloffs of Linkwitz-Riley 6th or 8th order filters to compensate a phase mismatch due to a driver offeset. One may think of this as a different technique since steeper rolloffs than a 4th order slope are used. But it is essentially the same thing. The only difference is the original rolloff rate before the relaxation applies. Technically, this should be viewed as "relaxed" LR6 or LR8 (not some different kind of LR4), because the resulting, total amount of phase mismatch at the xover point is 540 degrees (in the case of LR6; requires reversed tweeter polarity) or 720 degrees (in the case of LR8).
                      Last edited by jkim; 06-10-2008, 10:22 PM. Reason: Notes added and edited

                      Comment


                      • #12
                        Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                        It's a nice theory that explains everything.

                        I'm looking for the practical application of it in a filter that address the cone breakup of, say, a RS150, and brings it's frequency response to flat through the breakup region (and its phase curve too). At the moment I'm stuck with notch filters that suppress the driving signal at the breakup frequency and that mess up phase relationships in the rest of the crossover . . . the ones jeff was talking about above).

                        Something odd seems to happen with bsc as well . . . build it into a crossover, and then push the speaker back against the wall. Does phase at the driver change? What can I do, filter wise, to make it sound the same whether the speaker is out in the room or against the wall without changing frequency response or phase (in the crossover)?
                        "It suggests that there is something that is happening in the real system that is not quite captured in the models."

                        Comment


                        • #13
                          Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                          Originally posted by bmaupin View Post
                          Seems that the additional difference between the two methods is that the target is no longer LR4 @ 2kHz. The first method maintains the target and the second does not. I would not consider that the end of the world, but when I design a crossover I am trying to hit a target.

                          - Brad
                          Hi Brad,

                          The slightly lowered Fc in the sim of the second method is simply an artifact of this particular sim. If we want to, by adjusting the starting point of each rolloff, we can move it to the target Fc and still have the same effect.

                          -jAy

                          Comment


                          • #14
                            Re: Practical factors considered

                            Now, check out an example whose target response is 2100Hz LR4, with a 20mm offset using response plots provided by Jeff B for the RS150 and RS28.

                            These response plots follow the target LR4 responses quite closely through the XO region. They don't look much relaxed to me, yet we can all see the excellent on axis response as well as the tight phase alignment reflected in the 20dB reverse null.
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                            • #15
                              Re: Offsets, Asymmetrical Slopes, and Mysticism - revisited *PICS*

                              Originally posted by Deward Hastings View Post
                              It's a nice theory that explains everything.

                              I'm looking for the practical application of it in a filter that address the cone breakup of, say, a RS150, and brings it's frequency response to flat through the breakup region (and its phase curve too). At the moment I'm stuck with notch filters that suppress the driving signal at the breakup frequency and that mess up phase relationships in the rest of the crossover . . . the ones jeff was talking about above).

                              Something odd seems to happen with bsc as well . . . build it into a crossover, and then push the speaker back against the wall. Does phase at the driver change? What can I do, filter wise, to make it sound the same whether the speaker is out in the room or against the wall without changing frequency response or phase (in the crossover)?
                              There's only one other thing you can do and that is to treat the room. For placement against the wall, you'd need quite a bit of absorbing material to take care of boundary reinforcement. Even then, if the speaker has good bass extension, only ridiculous amounts of treatment would help.

                              If the response at the listening position is to remain the same, some form of EQ is required as the speaker is moved from a position radiating into 4pi space and towards 2pi space. If response changes, so does the phase since phase and amplitude are directly related.

                              And if you add damping to your notch filters by addition of a resistor, you can smooth out the breakup to make it more closely fit a smooth target response. That should take care of the phase problems you're citing.
                              R = h/(2*pi*m*c) and don't you forget it! || Periodic Table as redrawn by Marshall Freerks and Ignatius Schumacher || King Crimson Radio

                              Byzantium Project & Build Thread || MiniByzy Build Thread || 3 x Peerless 850439 HDS 3-way || 8" 2-way - RS28A/B&C8BG51


                              95% of Climate Models Agree: The Observations Must be Wrong
                              Gravity is an overrated force on the cosmic scale. Physicists are missing the bigger picture. They fell into a black hole and were never seen advancing the understanding of the cosmos again.

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