# Phase response formula for Butterworth filters?

Discussion in 'Electronic Design' started by Carlos, Dec 8, 2003.

1. ### CarlosGuest

Hi all,

I'm looking for the phase response formula of a Butterworth filter as
a function of its order and (f/fc), equivalent to the well known
amplitude one:

1
--------------------
SQRT(1+(f/fc)^(2*n))

If there isn't a single formula depending on n, I'd be grateful if you
could give me the formulas for 1st (I think that one is just
-Arctan(f/fc), right?), 2nd and 3rd order.

Carlos

Carlos, Dec 8, 2003

2. ### BaphometGuest

"Carlos" <> wrote in message
news:...
> Hi all,
>
> I'm looking for the phase response formula of a Butterworth filter as
> a function of its order and (f/fc), equivalent to the well known
> amplitude one:
>
> 1
> --------------------
> SQRT(1+(f/fc)^(2*n))
>
> If there isn't a single formula depending on n, I'd be grateful if you
> could give me the formulas for 1st (I think that one is just
> -Arctan(f/fc), right?), 2nd and 3rd order.
>
>
> Carlos

http://www.crbond.com/papers/btf.pdf

http://www.kemo.com/data/datasheets/responsecurves/fr_sfr.pdf

Baphomet, Dec 8, 2003

3. ### CarlosGuest

"Baphomet" <> wrote in message news:<>...
>
> Google [phase response butterworth filter]
>
> http://www.crbond.com/papers/btf.pdf
>

Thanks, I had actually found this one, but wasn't sure if I could just
replace W by (f/fc) and also wondered whether there would be a general
formula depending on N (which would be very convenient for the
crossover spreadsheet I'm working on). Anyway, I have plotted those
and they seem to match the curves shown in Dickason's Loudspeaker
Cookbook, so I'll go with them.

Thanks again

Carlos

Carlos, Dec 9, 2003
4. ### BanGuest

Carlos wrote:
|| Hi all,
||
|| I'm looking for the phase response formula of a Butterworth filter as
|| a function of its order and (f/fc), equivalent to the well known
|| amplitude one:
||
|| 1
|| --------------------
|| SQRT(1+(f/fc)^(2*n))
||
|| If there isn't a single formula depending on n, I'd be grateful if
|| you could give me the formulas for 1st (I think that one is just
|| -Arctan(f/fc), right?), 2nd and 3rd order.
||
||
|| Carlos

Carlos,
The phase response of a filter of arbitrary order and type is:

phi= -SUM arctan ai*(f/fc) / (1-bi(f/fc)^2)
i
with ai, bi being the coefficients of 2nd-order blocks.

--
ciao Ban
Bordighera, Italy
electronic hardware designer

Ban, Dec 9, 2003