What is the domain and codomain of a transfer function?

Question

Let's say I have the transfer function-

$\textbf{H}(j\omega)=\cfrac{1}{1+j\omega RC}$

Where does this function map to and from, and can it be plotted visually?

Answer 1

You are better off looking at the transfer function in the $s$-domain where it looks like $$ H(s) = \frac{1}{1+RCs} $$ From the Laplace transformation you know that $0\leq s<\infty$ and from the range of $s$ you get $0 < H \leq 1$. Your version of the transfer function is obtained when you use the substitution $s\mapsto j\omega$ which only works for a linear system (well, so do transfer functions themselves, but that's beside the point). This is essentially a rotation of $s$ onto the imaginary axis and renaming it frequency, and the magnitude of this complex function is the system gain as a function of the signal frequency.

How do you plot it visually? That's basically the majority of classical controller analysis in the frequency domain! Take a look at these two wikipedia pages for a start:

https://en.wikipedia.org/wiki/Bode_plot

https://en.wikipedia.org/wiki/Nyquist_plot

Bode and Nyquist plots are a staple of classical control and extremely useful in real controller designs. Also less frequently is used the Nichols plot

https://en.wikipedia.org/wiki/Nichols_plot