How Do You Design a Low Pass Filter Using a Butterworth Transfer Function?

[desmal]

I want to design LPF using Butterworth transfer function & I have the following information
Minimum gain attenuation Amin ,
Maximum gain attenuation Amax ,
Frequency of passband edge wp,
Frequency of stopband edge ws,
Amin,Amax,wp & ws can take any value

(Using only Butterworth table for wp=1 & є=1)=====>which is the proplem Here

 

[Corneo]

You have to calculuate the order of the Butterworth filter. Do you not have any equations or anything?

 

[desmal]

Hi Corneo

I don’t know if you understand me :-)

For example if I have Amax=1dB ,Amin=25dB ,wp=20π rad/s ,ws=30π rad/s & dc gain=1 V/V

I can find the Butterworth transfer function
T(jw)= (1+є² (w/wp)^2n)^-½ with є=0.5088 N=9
Then I can find the poles

But I want to get the poles from Butterworth table for є=1 & wp=1rad/s

n Factors of Polynomial T(s)
1 (s + 1)
2 s2 + 1.414s + 1
3 (s + 1)(s2 + s + 1)
4 (s2 + 0.7654s + 1)(s2 + 1.8478s + 1)
5 (s + 1)(s2 + 0.6180s + 1)(s2 + 1.6180s + 1)
6 (s2 + 0.5176s + 1)(s2 + 1.414s + 1)(s2 + 1.9318s + 1)
7 (s + 1)(s2 + 0.4450s + 1)(s2 + 1.247s + 1)(s2 + 1.8022s + 1)
8 (s2 + 0.3986s + 1)(s2 + 1.111s + 1)(s2 + 1.6630s + 1)(s2 + 1.9622s + 1)

For this table we know that Amax =3dB for all poles. This is mean even if we scale T(jw)--->T(jw/wp) in order to reach to wp we will still need Amax to be 1dB.

So how I can scale the transfer function & choose the order such that the new function will match my bounded value.

 

[Corneo]

I'm not sure if I fully understood your problem here. Are you trying to find the final Butterworth transfer function for any arbitrary $\omega_c$, given that you have found the normalized Butterworth transfer function?

 

[rbj]

For this table we know that Amax =3dB for all poles. This is mean even if we scale T(jw)--->T(jw/wp) in order to reach to wp we will still need Amax to be 1dB.

So how I can scale the transfer function & choose the order such that the new function will match my bounded value.

is one of your problems that you don't have butterworth polynomial for N=9? do you need a closed form expression?

also, what you should do with your spec is determine where the -3 dB frequency would be (assuming you did that right and correctly determined a 9^th order is needed.

 

[BobG]

You're looking for the middle term in your polynomials?

For your odds:

Divide 180 by n. (let it equal t)

Your first term is (s+1).
For the second term, in polar coordinates:
$$(1 \angle{t})+(1 \angle {-t})$$
Third polynomial:
$$(1 \angle{2t}) + (1 \angle {-2t})$$
etc.

For your fifth polynomial you have:
(s+1)
$$(1 \angle {36}) + (1 \angle {-36}) = 1.6180$$
The polynomial is $$s^2 + 1.6180s + 1$$
$$(1 \angle {72}) + (1 \angle {-72}) = .6180$$
The polynomial is $$s^2 + .6180s + 1$$

Making your final:
$$(s + 1) (s^2 + .6180s + 1) (s^2 + 1.6180s + 1)$$

etc.
Edit: Technically, you should use 180-t everywhere I used t, but, as you can see...

Edit: Here's a graphical representation of what you're doing: http://www.crbond.com/filters.htm