Constructing Bode Plots for \(H(s) = \frac{s}{s + 2}\)

[Dustinsfl]

I am trying to construct the Bode plots for \(H(s) = \frac{s}{s + 2}\).

For the magnitude Bode plot, \(H(i\omega) = \frac{1}{1 - i\frac{2}{\omega}}\) and the decibel gain expression is
\[
A_{\text{vdB}} = -10\log\bigg(1 + \frac{4}{\omega^2}\bigg).
\]
So this plots the first Bode plot, but I am not sure how to setup the phase/frequency Bode plots.

I have been following https://en.wikipedia.org/wiki/Bode_plot but I don't know what \(\omega_c\) is. Therefore, I can't \(\varphi\).

 

[jvicens]

Hello,

Thank you for reaching out. I can help you with constructing the Bode plots for \(H(s) = \frac{s}{s + 2}\).

To start, \(\omega_c\) represents the corner frequency, which is the frequency at which the magnitude of the transfer function is equal to \(\frac{1}{\sqrt{2}}\) or -3dB. In this case, \(\omega_c = 2\), which is the value of the denominator in the transfer function.

For the phase Bode plot, we can use the following expression:
\[
\phi = \tan^{-1}\bigg(-\frac{\omega}{2}\bigg)
\]
Note that this expression is valid for frequencies above \(\omega_c\). For frequencies below \(\omega_c\), the phase is constant at -90 degrees.

I hope this helps you construct the phase Bode plot for \(H(s) = \frac{s}{s + 2}\). Let me know if you have any further questions. Keep up the good work!