recreating in matlab Butterworth Filter filter response

Question

There is a manual which presents a filter response. In the video they present a formula and a plot of the response. However when I tried to implement it in MATLAB I get a totally different plot. Where did I go wrong implementing this formula? Thanks.

clc
clear all
s=0:0.01:50;
H=1./((s/20.01).^4+2.6131*(s/20.01).^3+3.4142*(s/20.01).^2+2.6131*(s/20.01)+1);
plot(s,20*log10(abs(H)))

https://www.youtube.com/watch?v=3yyp5JRqNXs&list=PLdciPPorsHunK-nh9PyXcA8rTy4m-TuOV&index=4

Answer 1

I was hesitant to answer your question because it shows that you need to review the basics. However, hopefully this will encourage you to do just that!

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The $s$ in $H(s)$ is a complex number $$s = \sigma + j\omega$$ where $\sigma$, the real part, represents a damping factor and $j\omega$, the imaginary part, represents an oscillatory component (for which $\omega$ is the angular frequency).

For sinusoidal steady-state analysis, we are interested in the behavior of the system at different frequencies but without the damping effect ($\sigma = 0$).

With that said, you only need minimal changes to your script:

omega = 0:0.01:50;
s = 1i * omega;