Bode Phase Plot

Question

I have a simple transfer function, given by:

$$ \text T(j\omega)=2\cdot(1-2\text{j}\cdot\omega) $$ I'm attempting to make it's Bode Phase plot without any calculator or program.

What i did:

$$ \text{Argument}T(j\omega)=\text{Argument}(2) + \text{Argument}(1-2\text{j}\cdot\omega) $$

$$ \text{Argument}T(j\omega)= \text{Atan}(\frac{0}{2})+\text{Atan}(\frac{2\cdot\omega}{1})+\pi $$

When $$\omega= 0 \Rightarrow \text{Argument}T(j\omega)=\pi $$

When $$\omega= infinity \Rightarrow \text{Argument}T(j\omega)=\frac{3\pi }{2} $$

Here is the solution in Matlab. Can some explain me how to use my calculations to obtain the same plot?

Thank you

Answer 1

Probably, $$\angle(1-2\omega \mathrm{j}) = -\operatorname{atan}\left(\frac{2\omega}{1}\right) \ne \operatorname{atan}\left(\frac{2\omega}{1}\right) + \pi.$$