Emitter resistance

The emitter resistance $r_e$ is the small-signal resistance seen looking into the emitter of a BJT (base held at AC ground, signal applied to the emitter):

$r_e=\frac{V_T}{I_E}\approx \frac{V_T}{I_C}=\frac{1}{g_m}$

The approximation uses $I_E\approx I_C$, which holds for any reasonable $\beta$ (they differ by the factor $\alpha =\beta /(\beta +1)\approx 1$). More precisely $r_e=\alpha /g_m$. It is the intrinsic emitter resistance of the device — not a physical resistor you add, but the small-signal resistance the emitter terminal presents.

Resistance at the emitter: $r_e=V_T/I_E\approx 1/g_m\approx 25\Omega$ at $I_C=1\text{mA}$ — much smaller than $r_{\pi }$. This is what the common-base configuration exploits.

Why it is β times smaller than r_π

This is the key intuition. The input resistance at the base is $r_{\pi }=\beta /g_m\approx 2.5\text{k}\Omega$ at 1 mA; the emitter resistance is $r_e\approx 1/g_m\approx 25\Omega$ at the same bias — a factor of $\beta$ smaller. The reason is which current flows through the controlling node:

• Drive the base: the base only carries $i_b=i_c/\beta$. The same voltage $v_{be}$ produces a tiny current, so the resistance looks large: $r_{\pi }=v_{be}/i_b=\beta /g_m$.
• Drive the emitter: the emitter carries the full current $i_e\approx i_c$ (β times bigger than $i_b$). The same $v_{be}$ now produces a large current, so the resistance looks small: $r_e=v_{be}/i_e\approx 1/g_m$.

Same junction, same $v_{be}$, but the emitter node sees β-times-more current than the base node, hence β-times-less resistance. Formally $r_e=r_{\pi }/(\beta +1)$ — the Resistance reflection rule in its purest form. Numerically, at $I_C=1\text{mA}$, $V_T=25\text{mV}$: $r_e=25\text{mV}/1\text{mA}=25\Omega$.

This very low emitter resistance is exactly what the Common-base amplifier exploits (its input resistance is just $r_e$, tens of ohms) and why an Emitter follower has such a low output resistance (≈ $r_e$). $r_e$ is the base–emitter element of the T-model form of the BJT small-signal model, the representation that is convenient whenever the emitter is the signal node.