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I have two signals

>> t = linspace(0, 10, 10000); % 0 to 10 seconds
>> u1 = 30*sin(2*pi*5*t); % Sine signal 1
>> u2 = 10*sin(2*pi*90*t); % Sine signal 2
>> u = u1 + u2; % Combine
>> spa(u, t); % Spectral analysis
>> grid on

And the spectral analysis shows these two signals u1 and u2

enter image description here

Now I want to filter the signal u by using a RC-filter

$$G(s) = \frac{1}{RCs + 1}$$

I say that at 90 Hz, I'm going to reduce the amplitude of the u2 sine signal with 50%, also half. 90 Hz is $s = 2\pi90$ radians.

So I put in the $s$ inside $G(s)$

$$G(2\pi90) = 0.5 = \frac{1}{RC2\pi90 + 1}$$

And now trying to solve $RC$

$$RC = \frac{1-0.5}{0.5*2\pi90} = 1.7684*10^{-3}$$

Assume that we are using $R = 1 kOhm$ and $C = 1.7684 uF$.

So I created my transfer function i MALTAB / Octave and plotted the bode diagram

>> R = 1000;
>> C = 1.7684e-06;
>> G = tf(1, [R*C 1]) % Transfer function
G =

  scalar structure containing the fields:

    num = 1
    den =

       1.7684e-03   1.0000e+00

    delay = 0
    tfnum =  1
    tfdash = ---------------
    tfden =  0.0017684s + 1
    type = TF
    sampleTime = 0

>> bode(G, 0, 600); % From 0 rad/s to a little bit over 2*pi*90 rad/s. As you can see, at ` 2*pi*90 = 565.49 rad/s`, I have `-3 dB`, that's `0.7079` in magnitude.

enter image description here

I do linear simulation of my transfer function and the spectral analysis. As you can see, why has not the amplitude of the peak of 90 Hz become half?

>> y = lsim(G, u, t);
>> spa(y, t); % Spectral analysis

enter image description here

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    $\begingroup$ For starters: $G(s)$ is a complex function that you need to evaluate on the imaginary axis, i.e. $s = j\omega$, $G(s) = \frac{1}{jRC\omega + 1)}$. If you want to cut the amplitude in half, you need solve for $|G(s)| = .5$, not $G(s) = 0.5$ . The more standard definition is the solve for $\omega = 1/RC$ which reduces the amplitude by $\sqrt{2}$ and the power by a factor of $2$ That's the -3dB point $\endgroup$
    – Hilmar
    Sep 7, 2022 at 20:43
  • $\begingroup$ @Hilmar I understand! I solved it now. Thank you so much. $\endgroup$
    – euraad
    Sep 7, 2022 at 21:20

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