I've got some doubts, too
Sallen Key filters are sometibes described with this transfer function:
1/(1+j*1/Q*(w/wc)-(w/wc)²)
with a butterworth filter's equation, I managed to find the same thing (and Q is 0.707, that seems correct)
but with a bessel, whos equation is
3/(3+3*j*(w/wc)+(w/wc)²) , it doesn't seem possible to have something like the first equation
neither with a chebyshev whos equation is
1/(1+e²(2*(w/wc)+1)²)
I tried to solve those equation all day long; now it's time to relax and listen a few cds
Sallen Key filters are sometibes described with this transfer function:
1/(1+j*1/Q*(w/wc)-(w/wc)²)
with a butterworth filter's equation, I managed to find the same thing (and Q is 0.707, that seems correct)
but with a bessel, whos equation is
3/(3+3*j*(w/wc)+(w/wc)²) , it doesn't seem possible to have something like the first equation
neither with a chebyshev whos equation is
1/(1+e²(2*(w/wc)+1)²)
I tried to solve those equation all day long; now it's time to relax and listen a few cds
Q = resonant magnification factor, i.e. the ratio of magnitude at resonance to that at infinty or zero (for high-pass and low-pass respecively).Svante said:
You're right, defining the filter responses by Q only works well at 2nd order. With higher orders, assuming they are constructed from multiple 2nd order sections, each section may have a different Q. Although the Q of each section is still fixed for each filter response, it's no longer the simplest way of describing them.
Ok, let's go through this.
-A first order filter is described by a f0 only. f0 is the frequency at which the asymtotes for very low and very high frequencies cross.
-A second order filter is described by A f0 and a Q. The definition of f0 is the same as for the first order filter. The definition of Q can be the amplitude at f0. Other definitions exist, but they give the same numerical value for a given filter.
-A third order filter is a first and a second order filter in series. This filter has two f0:s and one Q. This Q is not the amplitude at either f0, but the Q of the 2nd order section.
- A fourth order filter is two 2nd order filters in series. This filter has two f0:s and two Q:s. Neither Q is the amplitude at either f0, but the Q of the 2nd order sections.
-etc, etc...
The bottom line here is that many filters (almost all crossover filters) can be decomposed into a cascade of 1st and 2nd order filters. Each of these sections have a f0, and each 2nd order section also has a Q value. Trying to describe filters of orders >2 with a single Q value is doomed to fail, and will only add confusion. In the light of this:
All first order filters of a given f0 have identical transfer functions.
Filters of any order >1 are commonly described by a f0 (which may be different from those of the subsections) and a "family name", eg butterworth, bessel etc. The f0 can be defined in different ways, two common ways are by the -3 dB point, or the frequency at which the high and low frequency asymptotes cross. If the filter does not fit a well defined family, it is preferrably described by the f0:s and Q:s of all the subsections. Again, a single Q value will not tell the whole story except for the 2nd order filter.
Butterworth filters have sections with identical f0:s, but different Q:s. These filters are 3 dB down at the asymtotic f0.
Linkwitz-Riley filters are two cascaded butterworth filters and are therefore always of even order. These filters are 6 dB down at the asymtotic f0.
Bessel filters filters have cascaded sections with different F0:s and Q:s. These filters are typically more than 3 dB down at the asymtotic f0.
Tjebychev filters have cascaded sections with different F0:s and Q:s. These filters are designed to have a "ripple" of x dB in the pass band, and the f0 is usually defined as when the response goes below x dB. This f0 is lower than the asymtotic f0.
For crossover filters, Butterworth filters produce a flat sum if and only if the filter is odd order, Linkwitz-Riley filters produce a flat sum for even orders.
Bessel and Tjebychev filters do not produce a flat sum, so they are not principally interesting for crossover filters.
PS. When I speak of cascaded filter sections, I think of active filter sections, where there are no interaction effects between from loading the previous section.
HTH
-A first order filter is described by a f0 only. f0 is the frequency at which the asymtotes for very low and very high frequencies cross.
-A second order filter is described by A f0 and a Q. The definition of f0 is the same as for the first order filter. The definition of Q can be the amplitude at f0. Other definitions exist, but they give the same numerical value for a given filter.
-A third order filter is a first and a second order filter in series. This filter has two f0:s and one Q. This Q is not the amplitude at either f0, but the Q of the 2nd order section.
- A fourth order filter is two 2nd order filters in series. This filter has two f0:s and two Q:s. Neither Q is the amplitude at either f0, but the Q of the 2nd order sections.
-etc, etc...
The bottom line here is that many filters (almost all crossover filters) can be decomposed into a cascade of 1st and 2nd order filters. Each of these sections have a f0, and each 2nd order section also has a Q value. Trying to describe filters of orders >2 with a single Q value is doomed to fail, and will only add confusion. In the light of this:
All first order filters of a given f0 have identical transfer functions.
Filters of any order >1 are commonly described by a f0 (which may be different from those of the subsections) and a "family name", eg butterworth, bessel etc. The f0 can be defined in different ways, two common ways are by the -3 dB point, or the frequency at which the high and low frequency asymptotes cross. If the filter does not fit a well defined family, it is preferrably described by the f0:s and Q:s of all the subsections. Again, a single Q value will not tell the whole story except for the 2nd order filter.
Butterworth filters have sections with identical f0:s, but different Q:s. These filters are 3 dB down at the asymtotic f0.
Linkwitz-Riley filters are two cascaded butterworth filters and are therefore always of even order. These filters are 6 dB down at the asymtotic f0.
Bessel filters filters have cascaded sections with different F0:s and Q:s. These filters are typically more than 3 dB down at the asymtotic f0.
Tjebychev filters have cascaded sections with different F0:s and Q:s. These filters are designed to have a "ripple" of x dB in the pass band, and the f0 is usually defined as when the response goes below x dB. This f0 is lower than the asymtotic f0.
For crossover filters, Butterworth filters produce a flat sum if and only if the filter is odd order, Linkwitz-Riley filters produce a flat sum for even orders.
Bessel and Tjebychev filters do not produce a flat sum, so they are not principally interesting for crossover filters.
PS. When I speak of cascaded filter sections, I think of active filter sections, where there are no interaction effects between from loading the previous section.
HTH
phase_Accurate wrote:
Just one addition:
2nd order Linkwitz is actually a 2nd order Bessel (with both branches having the same POLE frequency) with one driver wired out of phase.
-------------------------------------------------
I don't think this is accurate. Bessel is a 2nd order TF with a Q of 0.577 and it is essentially a compromise between Butterworth and LR2. It has a compromise peak in amplitude to get a lower dip in power response.
Butterworth Q=0.707 is -3dB at Fc and has a 3dB peak in the vector sum when both sections are at the same frequency, but has flat power response.
LR2 is a Q of 0.5 and is -6dB at Fc and sums flat, but has a dip of 3dB in power response.
People still argue whether on axis frequency or power response is important. There are pros and cons, but on axis frequency response is more important unless the environment is very reverberant.
Getting hung up on alignments is not very useful, though. What we want is a flat sum and good off-axis behavior. Whatever gets you to that target on budget is what works.
Just one addition:
2nd order Linkwitz is actually a 2nd order Bessel (with both branches having the same POLE frequency) with one driver wired out of phase.
-------------------------------------------------
I don't think this is accurate. Bessel is a 2nd order TF with a Q of 0.577 and it is essentially a compromise between Butterworth and LR2. It has a compromise peak in amplitude to get a lower dip in power response.
Butterworth Q=0.707 is -3dB at Fc and has a 3dB peak in the vector sum when both sections are at the same frequency, but has flat power response.
LR2 is a Q of 0.5 and is -6dB at Fc and sums flat, but has a dip of 3dB in power response.
People still argue whether on axis frequency or power response is important. There are pros and cons, but on axis frequency response is more important unless the environment is very reverberant.
Getting hung up on alignments is not very useful, though. What we want is a flat sum and good off-axis behavior. Whatever gets you to that target on budget is what works.
I think I just have one thing to add to Svante's post
Svante, you say that defining a >2 order filter with only one Q is wrong, but I'm not sure about that.
Imagine a 4th order L-R. It's 2 Butterworth in series
a LR's Q is 0.5
a Butterworth's Q is 0.707
0.707*0.707=0.5
so I think you can also put a Q=1 and Q=0.5 in series, that will end in a LR4
But this is yet to confirm
Svante, you say that defining a >2 order filter with only one Q is wrong, but I'm not sure about that.
Imagine a 4th order L-R. It's 2 Butterworth in series
a LR's Q is 0.5
a Butterworth's Q is 0.707
0.707*0.707=0.5
so I think you can also put a Q=1 and Q=0.5 in series, that will end in a LR4
But this is yet to confirm
Ron E said:
I was just about to say something similar. I think your Q's are right, Ron, but that you can say nothing about the power response in general. The power response will depend on the distance between the drivers, in relation to the wavelength at the crossover frequency. What you can say, though is that LR with swapped polarity on one driver is the only configuration that produces flat on-axis response (for 2nd order, given ideal drivers)
Bricolo said:
1+0.5 will not yield the same transfer function as 0.7+0.7. It is true that the amplitude at fc will be the same, but to the sides of fc, the responses will differ. The bottom line is, the definition of Q is not the level at fc, unless the filter is 2nd order. So 1+0.5 is not LR.
4th-order LR is made of 2 cascaded 2nd-order Butterworth.
2nd-order LR is made of 2 cascaded 1st-order Butterworth.
Butterworth is the only kind of 1st-order filter you can get.
Thanks for bringing up the Q=1 + Q=0.5 example, I had always thought that would result in the same as 2 cascaded Butterworth's. Some playing with sims is in order.
2nd-order LR is made of 2 cascaded 1st-order Butterworth.
Butterworth is the only kind of 1st-order filter you can get.
Thanks for bringing up the Q=1 + Q=0.5 example, I had always thought that would result in the same as 2 cascaded Butterworth's. Some playing with sims is in order.
richie00boy said:
I'm not 100% sure about that.
I have the equations for a Nth order Bessel, Butterworth and Chebyshev
If I put N=1, and fc at -3dB, all the equations are equal
In other words, I wouldn't say that butterworth are the only 1st order filters, but that all 1st order filters (Bu, Be, Ch) are equal
Right, there is only one kind of first order filter for a given -3dB frequency. You may call it Butterworth, Chebychev or Bessel but not Linkwitz-Riley. The only thing you can change is the cutoff frequency.
BTW, two cascaded 1st order filters with the same fc is a second order filter with Q=0.5. But please, stop thinking that Q is the amplitude at fc, it is not except for the 2nd order filter. Doing so is only confusing IHMO.
BTW, two cascaded 1st order filters with the same fc is a second order filter with Q=0.5. But please, stop thinking that Q is the amplitude at fc, it is not except for the 2nd order filter. Doing so is only confusing IHMO.
OMG
Is there any english in this thread I'm looking at building or buying a 2way crossover for the adire extremis I will be getting and I'd like to gather some info on crossovers however this info is way over my head. Would it be better for someone without much education of electrical components just to buy a complete crossover kit or is there a faq or noobie site I can visit to pick up basics for crossover building?
Is there any english in this thread I'm looking at building or buying a 2way crossover for the adire extremis I will be getting and I'd like to gather some info on crossovers however this info is way over my head. Would it be better for someone without much education of electrical components just to buy a complete crossover kit or is there a faq or noobie site I can visit to pick up basics for crossover building?bser said:OMG
It can look a little overwhelming to start, eh? Try:
http://www.passivecrossovers.com
for a tutorial.
We were all n00bs once, so that's something to feel ok about. As for bolting on a crossover kit, that'd be like getting a set of rims on your car without knowing the hole pattern - it only fits by accident! You can either spent a fair bit of time coming up to speed on how to design crossovers, or you can find a kit designed around a specific set of speakers for which you build the cabinet. There's no shame in building a system like that - as a matter of fact some *really good* kits have been designed by well known top-notch engineers.
Francois.
Mathematical coincidence ?
Good question, Bricolo !
It is precisely the question I'm asking since some time, and I find nowhere the answer !
The problem is that the transfer function of any filter is the ratio of any two polynomials in the complex variable s=iw. Then you start reasoning about zeros and poles.
As an example, the transfer function of a low-pass filter is given by
f(s)=w0^2/(s^2+s*w0/Q+w0^2)
My real problem, and it seems that someone scratched it, that Bessel, Chebychev, ..., polynomial where defined long before that filters came into interest to humanity.
So ... was first the egg or the chicken ?
In other terms: filters where first defined and then someone said <Wow ! this is archetyped by a Bessel polynomial !>.
Or someone said: <Let's try with Bessel polynomials and then let us see what happens !>.
Yes, I know the mantra about <maximally flat, om, equi-ripple, om, sums flat, om, ...> but it seems to me that "mathematical coincidence" should not be a coincidence.
Good question, Bricolo !
It is precisely the question I'm asking since some time, and I find nowhere the answer !
The problem is that the transfer function of any filter is the ratio of any two polynomials in the complex variable s=iw. Then you start reasoning about zeros and poles.
As an example, the transfer function of a low-pass filter is given by
f(s)=w0^2/(s^2+s*w0/Q+w0^2)
My real problem, and it seems that someone scratched it, that Bessel, Chebychev, ..., polynomial where defined long before that filters came into interest to humanity.
So ... was first the egg or the chicken ?
In other terms: filters where first defined and then someone said <Wow ! this is archetyped by a Bessel polynomial !>.
Or someone said: <Let's try with Bessel polynomials and then let us see what happens !>.
Yes, I know the mantra about <maximally flat, om, equi-ripple, om, sums flat, om, ...> but it seems to me that "mathematical coincidence" should not be a coincidence.
Yep me too. Being the stubborn guy I am I had to build 'em in LspCAD and see for myself.
In the pic, the top one is a real 100Hz LR4 and the bottom one is a .5 + 1 kludge. They are pretty close but there are small differences of a dB or so in the shoulder region below Fc.
I had always thought the Q values Linkwitz picked for his stacked filters were arbitrary but I guess we'd better stick to the Q values on his web page for building LR filters of orders up to 10. Thanks for the heads up, guys.
http://www.linkwitzlab.com/filters.htm#2
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