Phase and argument of a complex signal

Question

How can I find the module and phase of $ \frac {1 + iwT }{101 - (2 T \pi f )^2 + 4 i \pi f T} $ ? I have to draw the graph of the module and the phase but I don’t understand an “ easy way “ to find them. I found that $ A_X (f) = \frac {\sqrt {1 + w^2 T^2 } }{ \sqrt {( 101 - (2 T \pi f )^2 )^2 + ( 4 \pi f T)^2 }} $ and that $ \phi_X (f)= arctg {1/wT} - arctg \frac{ 4\pi f t)} {101 - (2T \pi f )^2 } $ but I dint know if is correct and how to draw it

Answer 1

Hint:

The expression can be set in the form

$$\lambda\frac{1+ia}{(1+ib)(1+ic)}=\lambda\frac{\sqrt{1+a^2}}{\sqrt{1+b^2}\sqrt{1+c^2}}\text{cis}(\arctan a-\arctan b-\arctan c).$$

You can use a Bode plot. https://en.wikipedia.org/wiki/Bode_plot