Small-signal model

A biased transistor is a nonlinear device sitting at an operating point. Superimpose a SMALL signal on that operating point — small enough that the exponential curve looks straight over the excursion — and the device's response to that signal is linear. That means every tool from Circuit Analysis becomes available again for the signal, provided you work with deviations from the bias point rather than absolute values. This is linearisation, and it is the single most important technique in analogue electronics.

The key parameter is the slope of the exponential at the operating point: how much collector current a small change in base-emitter voltage produces. Differentiate $I_C = I_S e^{V_{BE}/V_T}$ and something remarkably clean falls out — the slope depends only on the bias current and the thermal voltage, not on the device at all.

With $V_T \approx 26$ mV at room temperature, a bias current of $1$ mA gives $g_m \approx 38$ mS — and any BJT biased at $1$ mA gives the same, regardless of part number, size, or β. That universality is why BJTs remain preferred for precision analogue work: their transconductance is set by a current you choose and a physical constant, while a MOSFET's depends on process and geometry.

The base, however, does draw current, so the model needs an input resistance. Since $i_B = i_C/\beta$ and $i_C = g_m v_{BE}$, the resistance looking into the base is: