output resistance of BJT

Question

I am working through Razavi mircroelectronics and regarding BJT resistances, there are three master rules:

Rule # 1: looking into the base, the impedance is r_π if emitter is (ac) grounded.

Rule # 2: looking into the collector, the impedance is r_ο if emitter is (ac) grounded.

Rule # 3: looking into the emitter, the impedance is 1/gm if base is (ac) grounded and Early effect is neglected.

But in the circuit below, the Resistance looking into emitter2 is show to be r_π || r_o || 1 / gm (and not just 1/gm).

Why is that?

^{simulate this circuit – Schematic created using CircuitLab}

Answer 1

It is not a surprise. Notice that in rule 3 you neglecting not only the Early effect (\$r_o\$).

But also a fact that:

\$\frac{1}{g_m}\$ << \$r_\pi\$

Aditional notice that

$$\frac{1}{g_m} || r_\pi = \frac{1}{ g_m + \frac{1}{r_{\pi}}} = \frac{r_{\pi}}{g_mr_{\pi} +1} $$

Also, did not forget that : \$ g_mr_{\pi} = \frac{I_C}{V_T}\frac{V_T}{I_B} = \frac{I_C}{I_B} = \beta\$

Therefore the impedance looking into the emitter is:

$$\frac{1}{g_m} || r_\pi = \frac{r_{\pi}}{\beta +1} = r_e $$

$$r_e = \frac{dV_{BE}}{dI_E} = \frac{V_T}{I_E} = \frac{r_\pi}{\beta +1} = \frac{\alpha}{g_m} = \frac{\beta}{g_m (\beta +1)} $$

But most a time we use a simplified approach thus, \$I_E \approx I_C\$ we can simplify our life and assumed that: $$r_e \approx \frac{1}{g_m}$$