Theory: Approximate Analysis

In this chapter, we will only consider operation at frequencies large enough where coupling capacitors C₁ and C₂ can be approximated to be short-circuits. Furthermore, we will not consider frequencies which so high that the intrinsic capacitances of the BJT itself affect circuit performance. These capacitances will be treated in another lab. Thus we will only be considering circuit performance for these midband frequences which for BJT circuits typically range from 10kHz to 1MHz.

Having defined our region of frequency operation, let's get back to amplifiers. As mentioned above the CE configuration is useful for amplifying small signal voltages. To understand this refer to Fig. 2.4 and the following discussion. Recall that a small change in base current leads to large change in collector current. Now, this small change in base current can be achieved by applying a small AC voltage to the base, which in turn will give rise to a large change in collector current so that $\Delta I_c=\beta\Delta I_b$.It follows that when a relatively large resistor R_C is placed between the power supply V_CC and the collector, the voltage variation across R_C and thus the voltage variation at the collector, due to the large change in collector current, will also be large. Thus, a small change in base voltage can lead to a large change in collector voltage. If we consider the input signal to be the change in base voltage, and the output signal to be the change in collector voltage, then the voltage amplification or gain will be $A_v=\frac{v_{out}}{v_{in}}~=~\frac{\Delta V_c}{\Delta V_b}$.By choosing the appropriate resistor values, we can design a simple CE amplifier with the voltage gain we want. To understand this, consider the following example. Recall, the voltage gain is $A_v=\frac{v_{out}}{v_{in}}~=~\frac{\Delta V_c}{\Delta V_b}$.The general procedure will be to find V_c and V_b from simple applications of Kirchoff's laws, and then find the small or incremental changes in these voltages due to an applied signal at the base.

Using KVL directly on the base emitter loop, and recalling that for most BJT's $I_e \approx I_c$, and thus substituting I_e for I_c, gives

V_b = I_c R_E + V_be

If we make an incremental change in V_b by applying a small signal to the base we obtain:  

$$\Delta V_b = \Delta I_cR_E + \Delta V_{be}$$ (28)

At this point we could continue our analysis in detail to determine $\Delta V_{be}$. However, it is useful to use our knowledge of diodes and BJT's to get some insight. Recall, the relationship between I_c and V_be is exponential. In other words, a small change in V_be leads to a large change in I_c. Furthermore, once a silicon diode turns on, its voltage drop will not change much from its DC value of $\approx 0.7V$. Thus $\Delta V_{be}$is almost always very small. Therefore, a zero order approximation can often be made to neglect $\Delta V_{be}$ compared with $\Delta I_cR_E$ to yield:

$$\Delta V_b \approx \Delta I_cR_E$$ (29)

Now let's look at the collector voltage V_C. From KVL we have

V_c = V_CC -I_cR_C

Since V_CC represents a DC power supply $\Delta V_{CC} = 0$, therefore making an incremental change in V_c leads to the following relationship:

$$\Delta V_c = - \Delta I_cR_C$$ (31)

Taking the ratio $\frac{v_{out}}{v_{in}}=\frac{\Delta V_{c}}{\Delta V_{b}}$ for the voltage gain yields:  

$$\frac{v_{out}}{v_{in}} \approx -\frac{R_C}{R_E}$$ (32)

Equation 2.18 shows that the approximate voltage gain of the CE amp is simply the ratio of the collector resistor to the emitter resistor. It's amazing how far you can get with this simple result.