Transistor Small-Signal Models — Worked Example

2N3904 CE stage at $I_C = 1\,\text{mA}$

Given: $I_C = 1\,\text{mA}$, $\beta = 150$, $V_A = 100\,\text{V}$, $V_T = 25\,\text{mV}$.

Find: $g_m$, $r_{\pi}$, $r_o$.

1. Transconductance.
   $$g_m = \frac{I_C}{V_T} = \frac{1\,\text{mA}}{25\,\text{mV}} = 40\,\text{mA/V}$$
   This is the slope of the $I_C$–$V_{BE}$ curve at the Q-point; it scales linearly with bias current.
2. Base-emitter resistance.
   $$r_{\pi} = \frac{\beta}{g_m} = \frac{150}{0.040\,\text{S}} = 3{,}750\,\Omega = 3.75\,\text{k}\Omega$$
   $r_{\pi}$ is the input resistance looking into the base in the small-signal model.
3. Output resistance (Early effect).
   $$r_o = \frac{V_A}{I_C} = \frac{100\,\text{V}}{1\,\text{mA}} = 100\,\text{k}\Omega$$
   A large $r_o$ means the transistor looks like a good current source; smaller $V_A$ (or higher $I_C$) degrades it.
4. Sanity check. The three parameters are consistent: $g_m \cdot r_{\pi} = 40 \times 10^{-3} \times 3750 = 150 = \beta$. This identity always holds and is a quick verification.