A few days ago I posted a webpage detailing my work to generate
equations for gain, phase, and group delay for a Butterworth lowpass filter,
using the basic polynomials. I could not find them anywhere on the Web or in
filter design books I own. The
only difference between calculating Butterworth lowpass, highpass, bandpass, and bandstop
filter values for gain, phase, and group delay
is how the relative frequency is defined. Simply substitute the following for ω
in any of the equations for gain, phase, or group delay. It's that simple. Graphs
are published below. Frequency units cancel out, so a 1 Hz cutoff plots the
same as a 1 kHz cutoff or a 1 GHz cutoff for gain and phase. The group
delay scale needs to be divided by a factor equal to the frequency units (÷10^{3}
for kHz, ÷10^{6} for MHz, etc.).
Butterworth Lowpass, Highpass, Bandpass, and Bandstop Filter Calculator with
Gain, Phase and Group Delay are now part of my free
RF Cafe Espresso Engineering Workbook™!
Lowpass
ω = ω_{n}/ω_{co}
Where ω_{n} is the frequency at which the equation is being evaluated,
and ω_{co} is the cutoff frequency

Highpass
ω = ω_{co}/ω_{n}
Where ω_{n} is the frequency at which the equation is being evaluated,
and ω_{co} is the cutoff frequency

Bandpass
ω = (ω_{n}/ω_{0}  ω_{0}/ω_{n})
/ BW
ω_{0} = sqrt (ω_{U} * ω_{L})
BW = (ω_{U}  ω_{L}) / ω_{0}
Where ω_{n} is the frequency at which the equation is being evaluated,
ω_{0} is the geometric average center frequency, ω_{U} is the upper
cutoff frequency, and ω_{L} is the lower cutoff frequency.

Bandstop
ω = BW / (ω_{n}/ω_{0}  ω_{0}/ω_{n})
ω_{0} = sqrt (ω_{U} * ω_{L})
BW = (ω_{U}  ω_{L}) / ω_{0}
Where ω_{n} is the frequency at which the equation is being evaluated,
ω_{0} is the geometric average center frequency, ω_{U} is the upper
cutoff frequency, and ω_{L} is the lower cutoff frequency.

I had
Archive.org save a copy of this page in order to prove this
is my original work.
Butterworth Highpass Filter
Gain
Phase
Group Delay

Butterworth Bandpass Filter
Gain
Phase
Group Delay

Butterworth Bandstop Filter
Gain
Phase
Group Delay

Posted December 13, 2023
