Butterworth filter design problems? Need your help

[binhjuventus]

I understand well the aspect mathematic of butterworth filter.

My problem now is, how to calculate the coefficients of the denominator of the transfer function in s-domain of an analog butterworth filter without looking up in the table? Of course, we are given a fomula to calculate the poles of a low pass filter?

In addition, How is the relationship between s-domain anf frequency domain?
Can I replace s = jω to give frequency response? or s=-jω? I can not distinguish between these 2 replacements...

That's because I intend to write a program in Mathematica to list these coeffients.

Looking forward to your helps...

 

[maziaar83]

denominator -----> Σ a(k)*s^k k=0~N

N: order of Butterworth
a(k) : coefficient of kth term

a(0)=1
a(k+1)=[cos((k*pi)/(2N)) / sin(((k+1)*pi)/(2N)) ] * a(k)

as you may noted the coefficients are symmetrical
for example for fifth order : a(5)=a(0) and a(4)=a(1) and a(3)=a(2)

you can write your program in mathematica using recursive loop

--------------------------------------------------------------------------------

put jω for s

but frequency domain formula is very simple
1 / (1+ε²ω^(2N))

if it helps push "Helped Me" button please

 

[Kral]

binhjuventus,
The poles of a Butterworth filter are symmetrical located on a circle of radius 1 for a cutoff frequency of 1rad/sec. Examples:
.
The poles of a 2-pole B filter are located at -0.5√2 + j0.5√2 And -j0.5√2 -j0.5√2
.
The poles of a 3-ple B filter are located at -1 and -0.5 +/- j0.5√3.
.
Linearly scale the poles to get a cutoff frequency other than 1rad/sec.
Regards,
Kral