MOSFET hybrid-pi model

The hybrid-π model is the default form of the MOSFET small-signal model: it draws the linearised transistor with the gate–source port left open and the controlled source plus output resistance hung across the drain–source port.

The circuit

• Gate to source: open circuit. No DC or AC gate current flows, because the gate is insulated by the Gate oxide. The full small-signal $v_{gs}$ appears across this open port and is the controlling variable.
• Drain to source: a voltage-controlled current source $g_m{v}_{gs}$, in parallel with the output resistance $r_o$. The current source is the linearised MOSFET square-law (slope $g_m$, the MOSFET transconductance); $r_o$ models Channel-length modulation (see MOSFET output resistance).

That is the entire model: $g_m$ and $r_o$, with an open input.

Why it is the natural choice for common-source

In a Common-source amplifier the input drives the gate and the output is taken at the drain, with the source at AC ground. The hybrid-π lines up perfectly with that topology: the input source connects straight across the open gate–source port (so the controlling $v_{gs}$ is just the input voltage), and the output port — drain to source — already has $g_m{v}_{gs}$ and $r_o$ in it. The gain falls out in one line:

$v_o=-(g_m{v}_{gs})(R_D\parallel r_o)\Rightarrow A_v=-g_m(R_D\parallel r_o)$

No rearrangement needed. For configurations where the source moves with the signal — the Common-gate amplifier in particular — the MOSFET T-model is usually tidier, but it is the same model: the hybrid-π and the T-model contain identical elements and always give the same answer. Use whichever makes the chosen topology least messy.