CE Amp with Emitter Bypass Capacitor

In electronics, it is often useful to bypass the emitter resistor with a capacitor as shown in Fig. 4.6.

  

Figure 4.6: CE Amp with Emitter Bypass Capacitor

This configuration is useful since at midband frequencies, C₃ acts to short the resistor R_E, thereby increasing the gain. But at DC, the presence of R_E facilitates achieving the desired DC bias point. The gain of the circuit in Fig. 4.6 is thus very similar to that in Fig. 4.3, except R_E is replaced with $Z_E=R_E\vert\vert\frac{1}{sC_E}$,and R_in is replaced with $Z_{in}= R_B\vert\vert(r_{\pi}+\beta Z_E)$.Thus, the expression for voltage gain of the emiter-bypassed CE amp is

$$\frac{v_{out}}{v_{in}}= \left(\frac{Z_{in}}{Z_{in}+\frac{1}... ...{g_m}}\right) \left(\frac{R_L}{R_{L}+R_C+\frac{1}{sC_2}}\right)$$ (76)

If the frequency increases towards the midband limit, then the impedances of the capacitors goes to zero and the expression for gain becomes

 

$$\frac{v_{out}}{v_{in}} \approx \frac{-g_mR_CR_L}{R_L+R_C}=-g_m(R_C\vert\vert R_L)$$ (77)

While the preceding expression is fairly simple, if we want to get a specific expression for the frequency-dependent gain of the CE amp in Fig.4.6, we have to put in explicit forms given above for Z_E and Z_in. Upon doing so we obtain the following fairly messy expression:  

$$\frac{V_{out}}{V_{in}}= \left(\frac{s^2R_Er_{\pi}C_1C_E + s(... ...g_m+1}\right) \left(\frac{R_L}{R_{L}+R_C+\frac{1}{sC_2}}\right)$$ (78)

It's interesting to note that when s becomes very large (4.13) reduces to (4.12), which is the expected result.