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Quality factor of higher order Butterworth LPF filter

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Junus2012

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Dear friends,

I have noticed from the table of Butterworth coefficients that Qtotal = Q1*Q2.

For example 2nd order Q = 0.707

4th order = 2nd order + 2nd order
Q1 = 0.54 Q2 = 1.3 Qtotal = 0.54*1.3 = 0.707

All filter manuals give the same standard coefficients for the individual stages

Is it possible to modify the coefficients of the individual but reaching the same final result, as an example, I choose Q = 0.707 for the first stage and Q = 1 for the second stage in the case of the 4th order filter and so on for the higher-order.

I have noticed this relationship only with Butterworth LPF

Thank you

Best Regards
 

Thank you dominik I have looked on the post, the man is surprised of concluding this magic fact as me :)
but my question is based on that magic :)
--- Updated ---

it really makes a difference

a 4th LPDfilter with Q1 = 0.707 and Q2 = 1 have less constraints than the same order with Q1= 0.54 and Q2 = 1.3
 

The simple answer is, you can modify the individual Q values, but the the filter is no longer of the Butterworth type. A Butterworth filter of specific order has unique pole parameters.
 
To achieve the best approximation of maximally flat the Q's are staggered and chosen to give a circular pole plot.
1599084017224.png



A Butterworth has circular equally spaced poles on a circle but nonlinear phase shift.

An elliptical filter describes the shape of the polar plot.

A Chebychev plot varies in eliptical ratio with steeper band stop, higher Qmax yet still terrible group delay.

A Bessel Filter gives the lowest Qmax and also the flattest group delay or most linear sloping phase.
1599085113222.png


Then there are many other combinations that optimize phase and steep skirts but as tradeoffs.
 
Dear FvM and Sunny

Thank you very much for your reply,

FvM confirms that individual order has unique Q, cant be changed,
Sunny says that might be other combination but with some tradeoff

If let may say a 4th LPF Butterworth with Q1 = 0.5412 and Q2 = 1.3065, those are the ideal standard polynomial.

You obviously know that in IC design we might not get such accurate value, since they are achieved by circuit components (in better case by ratios),
So let me say due to the imperfection, the Q1 became 0.5 then I raised Q2 to 1.414

Will you think that I will still have Butterworth approximation with this slight change?

Thank you once again
Regards
 

I'd expect that both Q and pole frequency undergo parameter variations. Thus you should compare the intended and the real filter characteristic and decide if the differences are acceptable..
 
Filter parameters change with component variations. That's why people usually define a filter transfer characteristic mask within which the actual transfer characteristic lies.
You are designing your 4th order filter as a cascade of two bi-quads with Q1=0.7 and Q2=1. How about Q1=1 and Q2=0.7? Have you thought about the pros and cons? For example, if you have your high Q bi-quad as last stage, because of the higher peaking from the higher Q it will result in more output noise, compared to the case when high Q stage is the put first.
 
sutapanaki said:
Filter parameters change with component variations. That's why people usually define a filter transfer characteristic mask within which the actual transfer characteristic lies.
You are designing your 4th order filter as a cascade of two bi-quads with Q1=0.7 and Q2=1. How about Q1=1 and Q2=0.7? Have you thought about the pros and cons? For example, if you have your high Q bi-quad as last stage, because of the higher peaking from the higher Q it will result in more output noise, compared to the case when high Q stage is the put first.
Click to expand...

Dear Suta,
In the procedure of design I put the lowest Q in the first stage and the higher next and next, in addition to what you explained if we put the higher Q in the first stage it might saturate the next stage, althaugh this is not the sensitive case with small Q filter like 4th Butterworth, it will be serious issue with high Q filter,

My concern we about to use Q = 0.707 and Q2 = 1 or Q1 = 0.5 and Q2 = 1.414 rather than the origional standard Q1= 0.54 and Q = 1.3,
 

Yes, if you put Q1=1 first it will shrink the dynamic range which is the thing that noise considerations trade off with. You will have to decide. Plus, it may saturate you second stage only if the spectrum of your input signal is uniformly distributed up to the corner frequency or is boosted at higher frequencies. If, however, the input spectrum is shaped with less content for high frequecies, perhaps it is ok then.

If you have a model of your opams and filter, you can quickly try Q1=0.5 and Q2=1.41 and see what it does with the filter shape.
 
Dear friends,

I have tried to build 4th-order, I used Q1 = 0.5 and Q2 = 1.4,

I could have successful Butterworth response,

By the way, generally speaking it is hard to reach fraction values in IC design like 0.54, etc,,, so I would say the approximation is working satsfactory

Thank you very much for your help in this regard
 

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