Butterworth Highpass & Bandpass Filter Gain, Phase, and Group Delay Equations

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 (÷10³ for kHz, ÷10⁶ 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/ω₀ - ω₀/ω_n) / BW

ω₀ = sqrt (ω_U * ω_L)

BW = (ω_U - ω_L) / ω₀

Where ω_n is the frequency at which the equation is being evaluated, ω₀ 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/ω₀ - ω₀/ω_n)

ω₀ = sqrt (ω_U * ω_L)

BW = (ω_U - ω_L) / ω₀

Where ω_n is the frequency at which the equation is being evaluated, ω₀ 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