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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 = ωnco

Where ωn is the frequency at which the equation is being evaluated, and ωco is the cutoff frequency

Highpass

ω = ωcon

Where ωn is the frequency at which the equation is being evaluated, and ωco is the cutoff frequency

Bandpass

ω = (ωn0 - ω0n) / BW

ω0 = sqrt (ωU * ωL)

BW = (ωU - ωL) / ω0

Where ωn is the frequency at which the equation is being evaluated, ω0 is the geometric average center frequency, ωU is the upper cutoff frequency, and ωL is the lower cutoff frequency.

Bandstop

ω = BW / (ωn0 - ω0n)

ω0 = sqrt (ωU * ωL)

BW = (ωU - ωL) / ω0

 

Where ωn is the frequency at which the equation is being evaluated, ω0 is the geometric average center frequency, ωU is the upper cutoff frequency, and ωL 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

Butterworth Highpass Filter Gain Plots (N=2-9) - RF Cafe

Gain

Butterworth Highpass Filter Phase Plots (N=2-9) - RF Cafe

Phase

Butterworth Highpass Filter Group Delay Plots (N=2-9) - RF Cafe

Group Delay

 

Butterworth Bandpass Filter

Butterworth Highpass Filter Phase Plots (N=2-9) - RF Cafe

Gain

Phase

Group Delay

 

Butterworth Bandstop Filter

Butterworth Highpass Filter Group Delay Plots (N=2-9) - RF Cafe

Gain

Phase

Group Delay

 

 

 

Posted December 13, 2023

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Copyright: 1996 - 2024

Webmaster:

    Kirt Blattenberger,

    BSEE - KB3UON

RF Cafe began life in 1996 as "RF Tools" in an AOL screen name web space totaling 2 MB. Its primary purpose was to provide me with ready access to commonly needed formulas and reference material while performing my work as an RF system and circuit design engineer. The World Wide Web (Internet) was largely an unknown entity at the time and bandwidth was a scarce commodity. Dial-up modems blazed along at 14.4 kbps while tying up your telephone line, and a nice lady's voice announced "You've Got Mail" when a new message arrived...

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