Input Resistance:

The input resistance is the resistance seen by the current source or voltage source which drives the circuit. For example, returning to the circuit in Fig. 2.7, the impedance seen by sinusoidal input signals looking into the capacitor is:

$$Z_{in}= \frac{1}{j\omega C_1} + R_B\vert\vert R_{TB}$$ (40)

Where R_TB is the resistance looking into the base of the transistor. As we have done throughout this lab, let's consider only cases where the frequency is large enough that the impedance of the capacitor can be ignored. We call these frequencies the midband range. At midband, the input impedance can be approximated as the following input resistance.

R_in= R_B||R_TB

R_B can be read directly from the circuit. However, we must determine R_TB. The resistance seen looking into the base can be determined by grounding the output, and then applying a small signal to the base $\Delta V_b$, and determining the base current that is drawn $\Delta I_b$. The ratio of the small signal base voltage to the small signal base current is then R_TB. To determine R_TB, let's first make the excellent approximation that I_c=I_e, and use Eqns. (2.14) and (2.23) to obtain:

$$\Delta V_b = \frac{\Delta I_c}{g_m} + \Delta I_c R_E$$ (42)

Now, recalling that $\Delta I_b=\frac{\Delta I_c}{\beta}$,and dividing $\Delta V_b$by $\Delta I_b$, we obtain  

$$R_{TB}= \frac{\Delta V_b}{\Delta I_b} = \frac{\beta}{g_m} + \beta R_E$$ (43)

Since $\frac{1}{g_m}$ is usually much smaller than R_E, when an emitter resistor is present a good approximation for the input resistance is simply $\beta R_E$. Of course, when the emitter is connected directly to ground, R_E=0 and under these circumstances $R_{TB}= \frac{\beta}{g_m}= r_{\pi}$ Note, that we have followed the customary convention and defined $\frac{\beta}{g_m}$ to be $r_{\pi}$.

It is interesting to observe that the resistance looking into the base is usually fairly large since it contains the multiplicative factor $\beta$. This can be explained because the current actually entering the base is I_b, while the current through R_E is approximately $\beta I_b$. This can alternatively be viewed as I_b going through an effective resistance of $\beta R_E$, thus giving the impression of a much larger resistance.