Considering the T model of MOSFET for small signals I want to obtain the input resistance from source to gate.
simulate this circuit – Schematic created using CircuitLab
The current flowing upwards through the $ 1/g_m $ resistance is
$$i_1 = displaystyle frac{V_{test}}{(1/g_m)} = g_m V_{test}$$
because the Gate G is connected to ground like the negative terminal of $ V_{test} $.
The current imposed by the voltage-controlled current source is
$$g_m v_{gs} = i_1 = – g_m V_{test}$$
(so it is flowing upwards too)
But this current source is short-circuited from ground to ground (because both Gate G and Drain D are connected to ground) and so it can create a current loop (with a current flowing between Gain, Drain and the current-source branch), with no current exiting from drain. This would not be correct for a MOSFET.
How is it possible to prove that the $ – g_m V_{test} $ current is flowing through the current-source branch and (through the drain) back to $ V_{test} $?
That is, how is it possible to prove that all the $ 1/g_m $ resistor current is absorbed by the current source and carried back to $ V_{test} $?
This is made to prove that the $ R_{in} $ seen from source is $ (1/g_m) || r_0 $.