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[SOLVED] Second-order Butterworth LPF transfer function

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STINGERX

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any order Butterworth LPF transfer function

hi
as you can see from the attached image to this post
i need to find out how to get the transfer function (depend on R,L,C)
from the butterworth polynomal formula
i need to know the mathematical procces
Untitled.jpg
 
Last edited:

i alredy know algebra :)
in a general case if for example i will need a 3rd order butterworth LPF i fill need to use the butterworth polynomal and figure out myself what to place there
not always i will be able to use the formulas you posted here

my question is about any order of butterworth filter
 

i already know that polynomal
i need to know how to find the values of R and L and C for any order of this LPF
or in other words what is the going on between the first equation and the second
 

Well you said you already know algebra and that's algebra. ;-)

the link you gave me explain how what is the transfer function of the filter in (s)
i have this formula to use with this polynom (s=p)
which give me the transfer function in terms of frequancy F and bandwidth B

and i need the transfer function to be in terms of R resistor L inducer and C capacitor
for any order of butterworth low pass filter
but i don't understand how

the image shown in my original post is for the second order lpf
 

SingerX,

at first, here are some basics on Butterworth approximation:

1.) For comparison, you need the n-th order lowpass function (normalization Ω=ω/ωc, ωc=3dB cut-off):

A(jΩ)=ao/[1+d1(jΩ)+d2(jΩ)²+d2(jΩ)³+...+]

2.) The MAGNITUDE function for a Butterworth approximation (n-th order) is

A(Ω)=Ao/SQRT[1+Ω^(2n)]

3.) Now you have to find the MAGNITUDE of the first expression and perform a comparison of the coefficients (d1...dn) ) with the second expression in 2.).
(That´s a nice task!!)

4.) Find the transfer function of the CIRCUIT you have in mind and compare again the coefficients of the corresponding parts in the denominator. This gives you parts values (R, L, C).
 
SingerX,

at first, here are some basics on Butterworth approximation:

1.) For comparison, you need the n-th order lowpass function (normalization Ω=ω/ωc, ωc=3dB cut-off):

A(jΩ)=ao/[1+d1(jΩ)+d2(jΩ)²+d2(jΩ)³+...+]

2.) The MAGNITUDE function for a Butterworth approximation (n-th order) is

A(Ω)=Ao/SQRT[1+Ω^(2n)]

3.) Now you have to find the MAGNITUDE of the first expression and perform a comparison of the coefficients (d1...dn) ) with the second expression in 2.).
(That´s a nice task!!)

4.) Find the transfer function of the CIRCUIT you have in mind and compare again the coefficients of the corresponding parts in the denominator. This gives you parts values (R, L, C).

Thank you!
 

4.) Find the transfer function of the CIRCUIT you have in mind and compare again the coefficients of the corresponding parts in the denominator. This gives you parts values (R, L, C).

For example, if you do this for 2nd order Butterworth, you arrive at d1=sqrt(2) and d2=1 (no surprise).
Now, these values have to be compared with the coefficients (expressed by parts values) of the corresponding circuit.
(Of course, both functions to be compared must appear in the same form with a "1" as a constant in the denominator).
 

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