Let me begin by saying there is nothing wrong with a Zobel impedance stabilization network and it always gives you the best and the most predictable results.
But there have been anecdotal reports of getting away with not using it. So I would like to explore if there might be a systematic way to design without Zobel.
Shown is an idealized representation of a woofer with its 2nd-order low pass crossover. It is oversimplified for 2 main reasons: 1, the impedance peak at resonance is completely ignored, and 2, the gradual impedance rise with frequency is not owing to voice coil inductance alone and as such it is not simple first order -6dB/octave, but approximately -3dB/octave or thereabouts depending on driver.
If the driver is purely resistive, or an appropriate impedance stabilization network has been applied, L1 and C1 could take on a wide range of values, keeping the crossover frequency fixed, corresponding to the wide range of possible alignments (LR, BW, Bessel, etc.)
Together with the inductive component L2 of the idealized driver it looks like a 3rd-order filter. No claims are made here that the driver's response corresponds to the response across R1 (far from it) or that the idealized model captures adequately what a real driver does. If anything both R1 and L2 varies with frequency. At certain frequencies the load isn't even inductive.
First observation: L2 and R1 can be (and should have been) measured, preferably mounted in the intended enclosure (cause the impedance certainly changes with the enclosure, although if the crossover frequency is high enough it makes little difference). If the impedance measurement includes phase measurement then L2 and R1 can be calculated from the impedance and the phase angle.
Second observation: if L1, C1 and L2 form a Butterworth filter there would be 3 equidistant poles on a perfect circle on the s-plane. Now we don't get to decide L2 - you can't have everything and if we are not keeping the impedance fixed something else will have to be variable and in this case it will be filter alignment.
So, there must be an ellipse on the s-plane, corresponding to a Chebyshev filter, and we can fix the 3 poles on the eclipse so that L2 equals to the value we measured, assuming L2 at the crossover point is the value to use. Butterworth is but one special case of Chebyshev filters just as a perfect circle is a special case of an ellipse.
I am working on the math to calculate L1 and C1 now and I will see if they simulate well in XSim.
But there have been anecdotal reports of getting away with not using it. So I would like to explore if there might be a systematic way to design without Zobel.
Shown is an idealized representation of a woofer with its 2nd-order low pass crossover. It is oversimplified for 2 main reasons: 1, the impedance peak at resonance is completely ignored, and 2, the gradual impedance rise with frequency is not owing to voice coil inductance alone and as such it is not simple first order -6dB/octave, but approximately -3dB/octave or thereabouts depending on driver.
If the driver is purely resistive, or an appropriate impedance stabilization network has been applied, L1 and C1 could take on a wide range of values, keeping the crossover frequency fixed, corresponding to the wide range of possible alignments (LR, BW, Bessel, etc.)
Together with the inductive component L2 of the idealized driver it looks like a 3rd-order filter. No claims are made here that the driver's response corresponds to the response across R1 (far from it) or that the idealized model captures adequately what a real driver does. If anything both R1 and L2 varies with frequency. At certain frequencies the load isn't even inductive.
First observation: L2 and R1 can be (and should have been) measured, preferably mounted in the intended enclosure (cause the impedance certainly changes with the enclosure, although if the crossover frequency is high enough it makes little difference). If the impedance measurement includes phase measurement then L2 and R1 can be calculated from the impedance and the phase angle.
Second observation: if L1, C1 and L2 form a Butterworth filter there would be 3 equidistant poles on a perfect circle on the s-plane. Now we don't get to decide L2 - you can't have everything and if we are not keeping the impedance fixed something else will have to be variable and in this case it will be filter alignment.
So, there must be an ellipse on the s-plane, corresponding to a Chebyshev filter, and we can fix the 3 poles on the eclipse so that L2 equals to the value we measured, assuming L2 at the crossover point is the value to use. Butterworth is but one special case of Chebyshev filters just as a perfect circle is a special case of an ellipse.
- No claims are being made that this methodology is any better than Zobel impedance stabilization. It is inferior except for component count.
- No claims are being made that this methodology will result in predictable response.
- Even if the response were predictable being a non-standard alignment we will have difficulty summing up flat with the high-pass section. Tweaking is still required.
- No claims are made that this replaces simulations. This just gives a starting point to begin with.
- The only claim is that there is one specific alignment that best cope with the inductance of the driver. The basis of this claim is that the poles needs to be evenly placed on an ellipse instead of being random dots which do not correspond to any geometric shape. Based on this we should be able to just calculate L1, C1 from L2, R1 and the xover frequency with no guesswork.
I am working on the math to calculate L1 and C1 now and I will see if they simulate well in XSim.
Attachments
Last edited:
Whether a zobel is needed has to do with the ideal vs actual electrical transfer function, as in this image from one of my blog posts:
The same filter is used for S1 and S2, but S2 has a Zobel applied.
So maybe you can take the approach of the intended -3dB , and the ideal vs. actual driver impedance at that location? The other thing I am thinking of is that if there is variation below -20 dB, we also don't care very much. For these reasons I'm tempted to imagine a "rule of thumb" that involves the LP filter Hz, and slope along with something about the driver's Z - R at the filter knee.
Feel free to grab the XSim files from there by the way too.
Best,
E
The same filter is used for S1 and S2, but S2 has a Zobel applied.
So maybe you can take the approach of the intended -3dB , and the ideal vs. actual driver impedance at that location? The other thing I am thinking of is that if there is variation below -20 dB, we also don't care very much. For these reasons I'm tempted to imagine a "rule of thumb" that involves the LP filter Hz, and slope along with something about the driver's Z - R at the filter knee.
Feel free to grab the XSim files from there by the way too.
Best,
E
Last edited:
In this example, I am aiming for a crossover point of 1.7kHz. S1 and S2 are the same woofer fed the same impedance data measured by REW. S2 uses L1 and C1 values from an online Linkwitz-Riley second-order calculator with the woofer's nominal impedance which is 4 ohm. S1 uses the Chebyshev method I outlined above. In both cases component values are rounded to the next nearest common value but this does not and should not make any noteworthy difference.
It is obvious from the graph that S2 ends up being way off its intended filter response. This demonstrates the perils of oversimplifying the driver to be resistive, using the nominal impedance for calculation but not applying an impedance stabilization network.
I'll try my best to outline the steps, as follows:
Discussion: there is a ripple in the filter response. If you had used a Zobel network and a traditional audio filter you would be able to get rid of the ripple. However, the LR2 network also had the ripple - the combined response of LR2 and the woofer's inductance does not look like LR2 at all, so the peak is not an issue of this method per se but an issue with not using impedance stabilization in general.
Both curves are steeper than 2nd-order.
It is obvious from the graph that S2 ends up being way off its intended filter response. This demonstrates the perils of oversimplifying the driver to be resistive, using the nominal impedance for calculation but not applying an impedance stabilization network.
I'll try my best to outline the steps, as follows:
- Determine the crossover point using these guiding principles: Tweeter playing low is desireable ... WHY ?
- Measure the impedance of your woofer in your enclosure. If you are new to this look into how REW does it.
- Find the row with your crossover frequency in the zma file. Convert the impedance and phase (i.e. polar coordinates) to real and imaginery (i.e. Cartesian coordinates) using cos and sin functions. The real part is resistive so it is your R1. The imaginary part turns into your inductance by this formula: omega = 2 pi f, and the imaginary part = j omega L (inductance).
- Use this online calculator used by the RF folks: RF Tools | LC Filters Design Tool
- Set the tool to 3rd order Chebyshev.
- Punch the resistive portion you calculated above as the output impedance. The input impedance shall be 0.
- Now, the Chebyshev filter frequency can be hard to choose, because if you enter your crossover frequency as is, you will NOT get -3dB or -6dB or anything reasonable at the crossover point. This is because Chebyshev is steeper than filters commonly used for audio. Multiply your crossover point by 0.7 as a starting point and enter that as the filter frequency.
- Adjust the filter frequency and the ripple. Aim for L3 of the online tool to match the impedance you calculated above from the imaginary part of your measurement. Look at the "Data Points" tab and your intended crossover frequency shall be somewhere between -3dB and -6dB.
- Transfer the calculated L1 and C2 values to your favourite crossover simulation tool and verify the filter response. You may have to tweak the values slightly.
Discussion: there is a ripple in the filter response. If you had used a Zobel network and a traditional audio filter you would be able to get rid of the ripple. However, the LR2 network also had the ripple - the combined response of LR2 and the woofer's inductance does not look like LR2 at all, so the peak is not an issue of this method per se but an issue with not using impedance stabilization in general.
Both curves are steeper than 2nd-order.
Attachments
Last edited:
Sometimes.. For example, the resonance peak of a compression tweeter is typically too difficult to work around while making alterations. I just fix the impedance and leave it. On the other hand the rising impedance of a woofer for a low pass filter is less complex and is often done either way.
I think to be truly "the best" we'd need to both stabilize the rising impedance (Zobel) and also neutralize the resonance impedance peak.
The problem with taking the mathematical approach is that you end up with a pointless result because:
-The inductive component of a woofer's impedance is highly lossy
-Cone breakup nodes cause the woofer's frequency response to peak up higher from the 1st order rolloff that the impedance would predict.
The best way to solve the problem is to just measure frequency response and impedance, put the measurements into a simulator, fudge the L and C of the filter until you meet an LR2 acoustical response. Even with a woofer that has little breakup, you will still need a zobel if you want to adhere to the correct slope for more than an octave or two above the corner frequency.
The only use of a filter calculator is to get some rough starting values. If i wanted to cross at 1.7kHz, i'd look up what the woofer impedance is at 1.7kHz and bang that into a filter calculator as the resistive impedance. Then adjust L and C up/down as required to meet the appropriate Q and Fc in the final acoustical response.
-The inductive component of a woofer's impedance is highly lossy
-Cone breakup nodes cause the woofer's frequency response to peak up higher from the 1st order rolloff that the impedance would predict.
The best way to solve the problem is to just measure frequency response and impedance, put the measurements into a simulator, fudge the L and C of the filter until you meet an LR2 acoustical response. Even with a woofer that has little breakup, you will still need a zobel if you want to adhere to the correct slope for more than an octave or two above the corner frequency.
The only use of a filter calculator is to get some rough starting values. If i wanted to cross at 1.7kHz, i'd look up what the woofer impedance is at 1.7kHz and bang that into a filter calculator as the resistive impedance. Then adjust L and C up/down as required to meet the appropriate Q and Fc in the final acoustical response.
Last edited:
The intent of the calculation is indeed to get a starting point. I did not claim any more than that.
I also never claimed that this in any way replaces measurements and experimentation. There is nothing exclusive between your methodology and mine. I did look very hard at the driver's measured response and phase and how that interacts with the filter. That I didn't talk about in this article didn't mean it was not done.
Now here is a question for you: you look at the measured FR of a combined electrical network/measured acoustic response. How do you know it is LR2, and you are so sure that it is not BW2, not Bessel, not Chebyshev? I would like to know how it is done, with supposed 100% certainty, that you can tell the Q by just looking at a graph of of the combined electrical/measured acoustic response.
I can't. If you could let me know how then great, I will have learned a way better methodology.
This whole thing came about after I have fudged a lot with L and C by trial and error and there are many combinations which look equally great in a plot, but the "knee" can be very different, with no guidance as to what would be good and why. It is possible that a true expert can look at 0's and 1's on a screen connected to Matrix and be able to decipher what is going on. The mortal without that tea leaf reading ability could use some mathematics for help.
This only gives the starting point with L and C, and I did still have to fudge after this, with the measured response.
At no point did I say any of this is necessary or beneficial, except for one case: if you are determined to not use a Zobel network my method will be better than ignoring the varying impedance. (Hint: read the title.) Any extrapolation beyond what I claimed as benefits are unwarranted. I hope by now you see where I am coming from.
I also never claimed that this in any way replaces measurements and experimentation. There is nothing exclusive between your methodology and mine. I did look very hard at the driver's measured response and phase and how that interacts with the filter. That I didn't talk about in this article didn't mean it was not done.
Now here is a question for you: you look at the measured FR of a combined electrical network/measured acoustic response. How do you know it is LR2, and you are so sure that it is not BW2, not Bessel, not Chebyshev? I would like to know how it is done, with supposed 100% certainty, that you can tell the Q by just looking at a graph of of the combined electrical/measured acoustic response.
I can't. If you could let me know how then great, I will have learned a way better methodology.
This whole thing came about after I have fudged a lot with L and C by trial and error and there are many combinations which look equally great in a plot, but the "knee" can be very different, with no guidance as to what would be good and why. It is possible that a true expert can look at 0's and 1's on a screen connected to Matrix and be able to decipher what is going on. The mortal without that tea leaf reading ability could use some mathematics for help.
This only gives the starting point with L and C, and I did still have to fudge after this, with the measured response.
At no point did I say any of this is necessary or beneficial, except for one case: if you are determined to not use a Zobel network my method will be better than ignoring the varying impedance. (Hint: read the title.) Any extrapolation beyond what I claimed as benefits are unwarranted. I hope by now you see where I am coming from.
Same as what I disclaimed near the beginning:
You and I were saying the exact same thing.
Another preemptive disclaimer related to this:
I knew one of you will say experimentation and simulation is paramount (that I never doubt) and calculation is of limited use (that I have problems with), so half of what I said were preemptive disclaimers. Please read.
Create an ideal RLC filter in your sim and overlay the ideal curve with the speakers acoustic one. That said, there really is no need to nail an ideal filter alignment in a speaker crossover because often you are making a compromise between frequency and phase responses between the two drivers you are crossing. Also getting the job done without using an excessive number of components, or excessively large/small values. If the tweeter has a little dip in the response near the XO frequency, you might allow the woofer to have a little peak to get the response flatter. Often it is just trail and error until you find something which solves the problem elegantly.
If your calculator requires Re and Le from the datasheet it will only be accurate to design crossovers at 1kHz because that is the usual frequency that manufacturers measure Le at. For frequencies other than 1kHz the effective Le will be different. A better way is to just look up the impedance plot to find what the approximate impedance magnitude at the frequency you wish to cross. Then just use any old LC filter calculator assuming it is a resistive load. You will probably still end up with a result like the ones in post #3 but it'll get the component values within an order of magnitude. The better speaker sims have optimisations functions too so you can just tell it to play with the values and hit a target crossover (but that is no fun 😛)
Last edited:
You must define "best" in measurable terms.
The art of engineering is not spending as much money as possible, but finding the most efficient solution for a given outcome. For modern SS amplifiers (A, A/B and D, etc.) I challenge you to simulate a meaningful, measurable reason for this with most 2-way speakers and an amplifier with a damping factor above 100 throughout the audio band. I think this is reasonable.
Flattening the impedance comes at the cost of more parts and significantly more power dissipation. Those parts must be big enough to handle that safely.
Even the simple RC Zobel used in a woofer is of vanishing value if the HP point is low enough OR if the final result meets the target curves. Sometimes you can compensate for the rising impedance in the filter sections itself. So again, I think the idea that Zobel's are always necessary, or necessary based purely on a driver spends money a little too enthusiastically.
Best,
E
@TMM I'm afraid you have completely misunderstood the process. Nowhere did I mention the datasheet. This is what I said:
And, also
So once again, you are saying (almost) the exact same thing as what I have already said:
And I did tell people to use an online Chebyshev calculator to find L and C, too.
There is only one very small difference: in my process the impedance (measured, can't be overemphasized) is separated into the resistive and inductive portions (urrgh... math). The resistive portion is used as the impedance to calculate the passive crossover which as we all know assumes a resistive load. The inductive portion is used to fixate the filter alignment to one of the well knowns. (3rd order Chebyshev, in this case)
On the whole, we could have just agreed to differ on this one. But in this specific design methodology the crossover alignment is important, and the gist of it is NOT purely theoretical, so I will have to explain.
There is only one crossover alignment for 1st-order. There is only one degree of freedom (i.e. one component value to choose).
For second order, the alignment can be described by a single variable, Q. We all know those Butterworth Q=0.707, LR=0.5, and so on. There are two degrees of freedom (choose 2 component values) so you get to choose both the crossover frequency and the Q.
For third order, and higher, there are 3 or more degrees of freedom (choose 3 component values). Given any random 3 component values, simply by picking 2 random inductors and 1 random capacitor, you will have "some" crossover point, no defined Q, and in all-likelihood, a wacky-looking filter response.
By letting the inherent inductance of the driver decide the Q, we are back to only one degree of freedom (crossover frequency).
Which is the whole point (and the only point) of my procedure.
And, also
So once again, you are saying (almost) the exact same thing as what I have already said:
And I did tell people to use an online Chebyshev calculator to find L and C, too.
There is only one very small difference: in my process the impedance (measured, can't be overemphasized) is separated into the resistive and inductive portions (urrgh... math). The resistive portion is used as the impedance to calculate the passive crossover which as we all know assumes a resistive load. The inductive portion is used to fixate the filter alignment to one of the well knowns. (3rd order Chebyshev, in this case)
On the whole, we could have just agreed to differ on this one. But in this specific design methodology the crossover alignment is important, and the gist of it is NOT purely theoretical, so I will have to explain.
There is only one crossover alignment for 1st-order. There is only one degree of freedom (i.e. one component value to choose).
For second order, the alignment can be described by a single variable, Q. We all know those Butterworth Q=0.707, LR=0.5, and so on. There are two degrees of freedom (choose 2 component values) so you get to choose both the crossover frequency and the Q.
For third order, and higher, there are 3 or more degrees of freedom (choose 3 component values). Given any random 3 component values, simply by picking 2 random inductors and 1 random capacitor, you will have "some" crossover point, no defined Q, and in all-likelihood, a wacky-looking filter response.
By letting the inherent inductance of the driver decide the Q, we are back to only one degree of freedom (crossover frequency).
Which is the whole point (and the only point) of my procedure.
My "religion" is math (as if that is not obvious enough). However, I am afraid of the other zealots. People who for example believe a full-range driver shall be Zobel'ed even when driven by an SS amp with high damping factor as you gave in your example. I tend to talk in ways that try not to offend anyone, even when mathematically some stuff does not hold any water. Math gets very little respect here, so my best course is to avoid any action with anyone, but simply grant them the benefit of doubt.
Within that back drop, "the best" is unlimited money, unlimited space and unlimited electricity.
However, I can perfectly see that a horn-loaded compression driver can get into trouble with response that requires notch filters to fix. Those work best when everything is properly stabilized. Those legitimately, rationally requires "the best" and that is not religious worship.
Ok, let me play the devil's advocate here, what if the high damping factor of a solid state amp is only obtained from heavy-handed negative feedback, and thereby it is not as immune to impedance fluctuations as the specs on paper would suggest?
🙂
You get the idea.
🙂
You get the idea.
Cyber,
Math is only worthwhile so long as the results create a meaningful outcome. Math is valueless. We assign value based on our own wants and needs. We can simulate the effect on FR based on the impedance of the amplifier and speakers, regardless of how the output impedance was achieved.
You may measure an 0.2 dB difference at 750 Hz. How much is that worth? That is where human value must make a choice.
For your own analogy, if you have unlimited money, you are no longer in an engineering discipline, you are now in a spending discipline and there is no reason to compare value to spend.
Best,
E
Math is only worthwhile so long as the results create a meaningful outcome. Math is valueless. We assign value based on our own wants and needs. We can simulate the effect on FR based on the impedance of the amplifier and speakers, regardless of how the output impedance was achieved.
You may measure an 0.2 dB difference at 750 Hz. How much is that worth? That is where human value must make a choice.
For your own analogy, if you have unlimited money, you are no longer in an engineering discipline, you are now in a spending discipline and there is no reason to compare value to spend.
Best,
E
For sure, math will have to be tempered by psychoacoustics. Otherwise, the reproduction of a square wave will become the next religion.
But nowadays we are so well cushioned from math that we forget it even exists. Everyone uses box simulation software and inside it is full of math pioneered by Thiele/Small. No one knows the math any more because it is all hidden in the software. But stop for a moment and appreciate the theoretical work that made our sport advance to where we are today - no longer are we designing ports by trial and error and anyone could design with just a spec sheet with reasonable predictability. This, in my view, is but one of numerous examples of what you referred to as "meaningful outcome".
You were suggesting that impedance fluctuations do not need to get stabilized with solid state amp which is of course true. But what were you basing your assessment on? Math. (Damping factor is the ratio between the input impedance of the speaker and the output impedance of the amp, to state the obvious.)
You use math to make rational choices. Everyone does. Why do we need to feel shame about so?
"The best" is for placating the zealots, when the full sentence always read like, "I knew it would have been the best because it would have measured a minuscule difference which would have been all but inaudible, but only if I had the money, space and time."
But nowadays we are so well cushioned from math that we forget it even exists. Everyone uses box simulation software and inside it is full of math pioneered by Thiele/Small. No one knows the math any more because it is all hidden in the software. But stop for a moment and appreciate the theoretical work that made our sport advance to where we are today - no longer are we designing ports by trial and error and anyone could design with just a spec sheet with reasonable predictability. This, in my view, is but one of numerous examples of what you referred to as "meaningful outcome".
You were suggesting that impedance fluctuations do not need to get stabilized with solid state amp which is of course true. But what were you basing your assessment on? Math. (Damping factor is the ratio between the input impedance of the speaker and the output impedance of the amp, to state the obvious.)
You use math to make rational choices. Everyone does. Why do we need to feel shame about so?
"The best" is for placating the zealots, when the full sentence always read like, "I knew it would have been the best because it would have measured a minuscule difference which would have been all but inaudible, but only if I had the money, space and time."
I think I will have to get back to the point Erik. In principle we are not in any way in disagreement: there is worthless math which give rise to inaudible results that we can safely ignore.
But to get back to the point, alignment is not pure math, it is audible. There is another thread running right now about the power response and on-axis response of LR vs Butterworth. Power response is clearly audible don't tell me otherwise. It is illuminating read and I learned a lot from that.
But to get back to the point, alignment is not pure math, it is audible. There is another thread running right now about the power response and on-axis response of LR vs Butterworth. Power response is clearly audible don't tell me otherwise. It is illuminating read and I learned a lot from that.
Last edited:
Note I didn't even have to invoke any concept that had dubious audibility like phase change/group delay.
It dawned on me that some of the words I spoke had potential of a major misunderstanding (and that huge misunderstanding may have already happened). When repeatedly challenged why I defined "the best" in a certain way, I answered that it was to placate the religious zealot. It suffices to say that that person has not appeared yet, and is not any of the participants of this thread.
The religious zealot is the "audiophile" who rely on measurements to the micro-ohm. This profile clearly did NOT fit any of you. I spoke of this imaginary person in a very derogatory manner. If there were a scale where one end was the irrational "audiophile" and the other end was rationality, I was closer to the "audiophile" end than any of you. I couldn't have been insulting myself so I could not have been insulting you.
Hope this clears things up. I apologize if anyone misunderstood and got offended. It was not my intent but I regret to have said what I said if you misunderstood and felt offended.
The religious zealot is the "audiophile" who rely on measurements to the micro-ohm. This profile clearly did NOT fit any of you. I spoke of this imaginary person in a very derogatory manner. If there were a scale where one end was the irrational "audiophile" and the other end was rationality, I was closer to the "audiophile" end than any of you. I couldn't have been insulting myself so I could not have been insulting you.
Hope this clears things up. I apologize if anyone misunderstood and got offended. It was not my intent but I regret to have said what I said if you misunderstood and felt offended.
It came to me, without surprise, that I am merely reinventing the wheel. Everything I did was itself a simplification of Wright's empirical model. There is a Wavecor document about it: http://www.wavecor.com/Transducer_equivalent_circuit.pdf
It expressly states that the Res and Les parameters on the spec sheet are usually unsuitable for crossover design, but the Wright model could.
It expressly states that the Res and Les parameters on the spec sheet are usually unsuitable for crossover design, but the Wright model could.
- Status
- Not open for further replies.