Q of 3rd order filters

Thread Starter

PeteHL

Joined Dec 17, 2014
363
@crutschow thank you for the filter design. Briefly just now I looked at the tool that you gave a link to in your post #7. That looks interesting, but the problem is that I think connecting the two filters in my first post in cascade to form a band-pass filter, and the cut-off frequency of the high-pass filter at the middle of the pass-band (566 Hz) seems to eliminate audible ringing. It seems that making one filter 2nd order and the other 3rd order, in forming a band-pass filter, makes the chore easier.

It may be that I will still have to revise the filter, in which case thank you for the link to the tool.

-Pete

crutschow

Joined Mar 14, 2008
27,392
It seems that making one filter 2nd order and the other 3rd order, in forming a band-pass filter, makes the chore easier.
Why do you think that?

MrAl

Joined Jun 17, 2014
8,347
Hello,

In second order filters the Q is directly related to the damping factor. I think it is inversely proportional.
In 3rd order or higher though we can have different responses so i think it depends on the particular filter because we have a 'dominate' response that takes over which can mimic a second order but not exactly.

For a 3rd order bandpass, we might see a wide stretch mid band where the gain is almost the same and at either end we see a roll off. The simple way to calculate Q then is the simple:
Q=Frequency/Bandwidth
where the bandwidth is the frequency difference between the points where the amplitude is 3db down. The 'Frequency' in that formula is usually taken to be the very center frequency:
Fc=(FH+FL)/2
where FH is the upper -3db frequency point and FL is the lower -3db point.
So if we wanted to state it in these terms we would have:
Q=((FH+FL)/2)/(FH-FL)=(FL+FH)/(2*(FH-FL))

So it is quite easy to calculate in theory although in practice we might have to solve a 6th order equation for a 3rd order filter in order to get the quantities required.

But there are some interesting questions that come up. In a 3rd order bandpass filter the response in the passband may not be exactly perfectly flat. There may be a peaked point and it may not be in the middle. The question is what to do then.
My suggestion then is to stick with the previous formula unless the peaked point is much much higher than the other parts, then you have to make s decision based on the application. If the peaked part is significant, you may have to use that as the center frequency, if it is not impordtant then use the previous formula, if it is of moderate importance then you might use the average value response to determine the center frequency.

I'd be interested to hear other ideas on this too and BTW we looked at such a response here in this forum not too long ago (a 3rd order bandpass with a slight peakedness to one side of the passband) although i am not sure if i can find it again. It may be in the Homework section, but we can look at other responses too.

@LvW it would be interesting to hear your ideas on this. I agree that the roll off is related more generally to the order except in 2nd order filters where it is directly related to the Q as well as the order.

MrAl

Joined Jun 17, 2014
8,347
Attached to this post is a LTspice simulation of a high-pass (Fc = 400 Hz) and a low-pass (Fc = 800 Hz) 3rd-order active filters with op amps. My question is whether or not it is possible to calculate Q of the filters based on the values of resistance and capacitance of the filters.

Based on what is advised in the book Op Amps for Everyone by Mancini, as I am interested in getting as low of a Q as possible, I simulated a Bessel 3rd order unity gain high-pass filter and found that with respect to frequency one octave below the cut-off frequency, attenuation is only -12 dB when it should be -18 dB given that it is supposed to be 3rd order. I would prefer the design from Mancini as I know in advance that it has the lowest possible Q, but it also needs to truly be a 3rd order filter.

If anyone knows the answer to this I will be appreciative.

Thanks in advance,
Pete
I would guess offhand this would be a 3rd order passive RC filter but maybe you can show the formula and circuit you talk about and what you want to use this for. Often the application is important when we get into more advanced ideas about circuits because many generalizations are directed more at 2nd order filters because they are so widespread and important, and small deviations from the norm may make a big difference in the application.

Papabravo

Joined Feb 24, 2006
16,459
Below is the LTspice simulation of a single opamp 3-pole Bessel LP filter with a -3dB point of 800Hz using the tool I referenced in post #7 (although I had to set the characteristic frequency to 440Hz to get the -3dB 800Hz rolloff).

Notice that the pulse transient response has very little overshoot (likely inaudible).

You can use the tool to similarly design your 400Hz high-pass filter.

View attachment 204275
View attachment 204276
Two questions on the schematic:
1. Can you explain how you get to use an LMC6484 model with a "U" designator, as opposed to a subcircuit with a "X" designator?
2. What is the meaning and purpose of the small degree symbol on the resistors?
Thanks.

PS I tried this in a PM, but apparently that is not possible.

crutschow

Joined Mar 14, 2008
27,392
Can you explain how you get to use an LMC6484 model with a "U" designator, as opposed to a subcircuit with a "X" designator?
I put the op amp model file in the library and then modified an op amp .asy file with a changed name and referencing the model so it appears just like any other op amp in the library.
What is the meaning and purpose of the small degree symbol on the resistors?
I modified the resistor icon to indicate the current-flow direction (positive flow is from bubble end to other end).

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Papabravo

Joined Feb 24, 2006
16,459
I put the op amp model file in the library and then modified an op amp .asy file with a changed name and referencing the model so it appears just like any other op amp in the library.
I modified the resistor icon to indicate the positive end of the resistor and the current-flow direction.
I thought the models in the library were binary. Maybe that is only for those models where the manufacturer wants to keep their information proprietary.

Thread Starter

PeteHL

Joined Dec 17, 2014
363
Why do you think that?
If both filters forming a band-pass filter are 2nd order, and the cut-off frequencies of the filters are 1 octave or less apart, then relative phase of the output of each filter is tending to be 180 degrees in the pass-band. If one filter is 2nd order and the other is 3rd order, then that possibility of negating when the the cut-off frequencies are close together is not the case.

Let's say that this my hypothesis as I haven't tried simulating this. Also I don't have a mathematical understanding of how phase combines when the cut-off frequencies are close together. My opinion is based on apparently getting a more satisfactory result in configuring a band-pass filter with a pass-band of about one octave from a 2nd order high-pass and 3rd order low-pass than where both filters are 2nd order.

I could be correct, or maybe not.

-Pete

crutschow

Joined Mar 14, 2008
27,392
I thought the models in the library were binary.
All the LTspice ones I'm aware of are ASCII.
I think a lot of PSpice models are binary.

Papabravo

Joined Feb 24, 2006
16,459
All the LTspice ones I'm aware of are ASCII.
I think a lot of PSpice models are binary.
I did figure that out by looking at a larger sample of parts in the existing libraries and doing some correspondence checks between models and symbols. Thanks for pointing this out. For some reason, from back in the Switcher Cad III days, I thought all sub circuits had to use the "X" designator.

LvW

Joined Jun 13, 2013
1,278
@LvW it would be interesting to hear your ideas on this. I agree that the roll off is related more generally to the order except in 2nd order filters where it is directly related to the Q as well as the order.
I must admit....I am not able to give any substantial comment or recommendation - unless the questioner gives some clear requirements or specifications. From the beginning I had the impression that he needs TWO DIFFERENT filters (3rd order). Now (post#28) he, suddenly, speaks about a 2nd-order bandpass. Perhaps he is not aware of the fact that in this case the slope of the attenuation outside the passband is of 1st order only...?

Thread Starter

PeteHL

Joined Dec 17, 2014
363

Papabravo

Joined Feb 24, 2006
16,459
@MrAl The 3rd order Bessel high pass filter in Mancini's book is on page 310 of the pdf. The design according to this book of an identical filter but low-pass is a little unclear to me. Here is the link:

https://web.mit.edu/6.101/www/reference/op_amps_everyone.pdf

Regards,
Pete
To change a highpass to a lowpass or vice versa you swap the R's and the C's. The calculations remain the same. Same for the first order section. Of course you are allowed to change the values, within the constraints, to make the selection of values rational. For example in the highpass you might select the C's to be 1nF and calculate the R's. In the equivalent lowpass you might pick the R's to be 10KΩ and compute the C's.

Refer back to p. 305 , §16.4, Figure 16–23. Low-Pass to High-Pass Transition Through Components Exchange

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crutschow

Joined Mar 14, 2008
27,392
I thought all sub circuits had to use the "X" designator.
True if it's a general sub circuit.
The op amp symbols use the U reference, of course.
Shown below is the .asy file for the LMC6384A with the corresponding attributes which references the model file (either .lib or .sub).
I just modified and saved this from an op amp already in the library.
You right-click on each of the terminals to make sure they are referenced to the proper sequence in the model file.

Papabravo

Joined Feb 24, 2006
16,459
True if it's a general sub circuit.
The op amp symbols use the U reference, of course.
Shown below is the .asy file for the LMC6384A with the corresponding attributes which references the model file (either .lib or .sub).
I just modified and saved this from an op amp already in the library.
You right-click on each of the terminals to make sure they are referenced to the proper sequence in the model file.

View attachment 204473
View attachment 204475
Thanks for that additional info. My temporary brain inhibition was looking at one or two Analog Devices files and finding them in binary format. I jumped to the erroneous conclusion that they were all like that.

One more thing. As your are no doubt aware the interwebs has collected numerous "SPICE" models over the years of the same component from various manufacturers. I have four of them for the LMC6484. How would you select the one you wanted to use. Is there any way to vet the various models, short of comparing them side by side?

Thread Starter

PeteHL

Joined Dec 17, 2014
363
To change a highpass to a lowpass or vice versa you swap the R's and the C's. The calculations remain the same. Same for the first order section. Of course you are allowed to change the values, within the constraints, to make the selection of values rational. For example in the highpass you might select the C's to be 1nF and calculate the R's. In the equivalent lowpass you might pick the R's to be 10KΩ and compute the C's.

Refer back to p. 305 , §16.4, Figure 16–23. Low-Pass to High-Pass Transition Through Components Exchange
Thank you, that is very helpful for me.
-Pete

Thread Starter

PeteHL

Joined Dec 17, 2014
363
So which filter is it?
Example 16-4 on p.310 and Fig. 16-30 on p.311. Those pages are of the *.pdf.

Papabravo

Joined Feb 24, 2006
16,459
Example 16-4 on p.310 and Fig. 16-30 on p.311. Those pages are of the *.pdf.
I did some additional reading of Chapter 16 on Active Filter Design and tried to look at using their tables for a Bessel filter. There is what is perhaps a minor flaw in the approach. The whole point of a Bessel filter is to provide linear phase in the passband which produces a constant group delay. The formulation of the second order transfer function has a 1 for the constant term, and their Bessel filters cannot maximize the width of the passband over which the group delay is constant. Van Valenburg shows in Analog Filter Design, that for the 2nd order transfer function to do this it must have the same coefficient for the constant and the linear term. Their Bessel Filters do the best they can with the constraints, but they are not optimal. The poles are outside the unit circle and that is a good thing.

In order to compensate for this, I suppose you could always revise your choice of ω₀. Perhaps this was related to @crutchow having to change the corner from 440Hz to 800Hz when he played with the online calculator.

EDIT: I now have a clearer picture of what is happening. Turns out there are two ways to begin the design a Bessel Filter. One of the starting points is to begin with a transfer function that is normalized for a unit Group Delay. This is the exposition in the text by Van Valkenburg. The other method is to begin with a transfer function that is normalized for a unit radian frequency. This is the method used by Mancini. Both methods produce viable filters, only the starting points differ. When you use a filter calculator (online or in a standalone tool) it might be a good idea to be aware of which one you are getting.

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