BJT hybrid-pi model

The hybrid-π model is the most common form of the BJT small-signal model. It represents an active-mode BJT as three elements between its base, emitter, and collector terminals:

• An input resistance $r_{\pi }$ connected between base and emitter — the BJT input resistance, $r_{\pi }=\beta /g_m$. This is where the small-signal base current $i_b$ flows, and the controlling voltage $v_{be}$ appears across it.

• A controlled current source between collector and emitter representing the amplifying action. It has two equivalent expressions:

  • a voltage-controlled current source $g_m{v}_{be}$ (controlled by the voltage across $r_{\pi }$), or
  • a current-controlled current source $\beta i_b$ (controlled by the current through $r_{\pi }$).

  These are the same source: since $v_{be}=i_b{r}_{\pi }$ and $r_{\pi }=\beta /g_m$, $g_m{v}_{be}=g_m{i}_b{r}_{\pi }=g_m{i}_b(\beta /g_m)=\beta i_b$.

• An optional output resistance $r_o=V_A/I_C$ in parallel with the source, collector to emitter, included only when the Early effect matters (e.g. computing maximum gain or output resistance).

The hybrid-π model in its two equivalent forms: a VCCS $g_m{v}_{be}$ (left) or a CCCS $\beta i_b$ (right); both have $r_{\pi }$ between base and emitter. The bottom panel adds $r_o$ between collector and emitter for the Early effect.

The hybrid-π is the natural choice whenever the base is the input node, because $r_{\pi }$ sits right there as the input resistance — so it is what you reach for in Common-emitter amplifier and Emitter follower analysis. When the signal instead drives the emitter (common-base, emitter-follower output resistance), the algebra is cleaner with the BJT T-model, which is the exact same model rearranged so that $r_e$ rather than $r_{\pi }$ sits between base and emitter. Both forms are interchangeable and always give the same numerical answer.