At the university we have seen this circuit as a Phase Modulator:
simulate this circuit – Schematic created using CircuitLab
where \$m(t)\$ is the message at audio frequency and \$v_p(t)\$ is the carrier at radio frequency.
The idea is that \$m(t)\$ change the value of \$g_m\$, modulating the phase of \$v_p(t)\$.
The proposed small signal linear circuit is this:
The first thing I do not understand is how the bypass capacity for \$ m (t)\$ is a short-circuit at audio frequencies and an open-circuit at radio frequency, as can be seen from the proposed small signal scheme. Also why do you think the prof used this model for JFET? It is an extremely approximate model for very low frequencies, it is not good especially if I work at slightly higher carrying frequencies.
\$g_m\$ depends on \$ m(t) \$, so it is a function of time. In the frequency domain then \$ g_m \$ is a function of \$ \omega \$ and i cannot consider it as a number, which has been done both on the book and the prof, obtaining the following expression (approx):
$$\frac{v_{out}\left( \omega \right)}{v_{p}\left( \omega \right)}=\frac{1-\frac{g_{m}}{j\omega {C}}}{1+\frac{g_{m}}{j\omega {C}}} $$
How can this analysis be correct even if in this case \$ g_m=g_m(t) \$?
Thank you.
PS: Sorry if my English is not very well.
