A bipolar junction transistor uses coupled PN junctions to control collector current via base-emitter injection. In forward active mode, the collector current is approximately $I_C \approx \beta I_B$ and more fundamentally $I_C \approx I_S e^{V_{BE}/V_T}$. The exponential transconductance is one reason BJTs remain valuable in precision analog design.

Linearizing around a bias point yields the hybrid-$\pi$ model, where $g_m = I_C/V_T$, $r_\pi = \beta/g_m$, and $r_o \approx V_A/I_C$ due to the Early effect. These parameters connect device physics to gain, input impedance, and output resistance in amplifier stages.

Bias design must stabilize the operating point against temperature drift, parameter spread, and loading. Common techniques include emitter degeneration, voltage-divider bias, and current mirrors. Robust biasing is essential because $V_{BE}$ and $I_S$ vary strongly with temperature.