Nth order Butterworth filter transfer function for a given roll-off frequency

I want to plot a fourth order low pass Butterworth for any given cut off freq. But I'm lacking the correct transfer function. I saw some polynomial transfer functions in s domain. But their cutoff is 1rad/s. Here is a table I found for wc=1rad/s:

Is there a quick way to modify for example the 4th order filter transfer function above for a different wc other than 1rad/s? Is there a quick way to obtain or is there a lookup table designated for filter transfer functions?

The given set of functions is normalized to wc=1rad/sec. That means: The frequency variables in all expressions are to be interpreted as S=s/wc (capital letter S).

Therefore, you have nothing to do than to select different cut-off frequencies wc and to rescale the variable. Example: For wc=10rad/ses you must set S=s/10.

• I was surprised to learn by simulation that the Butterworth filter not necessarily critically damped or N>2 inspite of maximally flat, higher orders have overshoot and ringing on a step response yet critically damped for 2nd order. TY Mar 20, 2019 at 20:58
• @SunnyskyguyEE75 You must have made a mistake along the way because the 2nd order Butterworth has overshoot and ringing, too. Apr 14, 2019 at 13:25
• TY @aconcernedcitizen 2nd order ~ 4.1% overshoot no wonder I never liked Butterworth filters. vs Bessel for best overshoot + phase response Apr 14, 2019 at 14:40

A more precise expression of Butterworth Polynomials where you divide s by any constant to match your desired frequency.

Just a different way of expressing the polynomials

To illustrate what the two gentlemen already answered, a quick plot can help. Below are transfer functions in which the crossover frequency is passed as a parameter for a 2nd-order and higher-order expressions. The selected frequency is 10 Hz as an example.