What is the ratio of a transfer function $G(s)$?

Question

If I have a transfer function $G(s) = \frac{Y(s)}{U(s)}$ where the $G(s)$ is the ratio between the amplitude of $Y(s)$ and $U(s)$. In what unit is the ratio?

Let's say that we have a input signal $u(t) = a_isin(2\pi\omega_it)$ and a output signal $y(t) = b_isin(2\pi\omega_it)$.

Does this mean that the amplitude ratio $G(s)$ is $\frac{b_i}{a_i}$ at $s = 2\pi\omega_i$, where $i = 0, 1, 2, 3, 4, \dots , n$

Answer 1

There is no unit or, in other words, the unit is irrelevant.

The output signal $y_i$ of the stable system $G$ for the input $u_i(t)=a_i\sin(\omega_i t-\psi_i)$, $a_i,\omega_i>0$, $\psi_i\in[0,2\pi]$, for some $i=1,\ldots,N$, is given by

$$y_i(t)=b_i\sin(\omega_i t-\psi_i+\phi_i), b_i>0,i=1,\ldots,N$$

where $|G(j\omega_i)|=b_i/a_i$ and $\phi_i=\arg(G(j\omega_i))$. This also means that

$$G(j\omega_i)=\dfrac{b_i}{a_i}e^{j\phi_i},i=1,\ldots,N$$