C.3. Amplifier Voltage Gain

Any transistor amplifier stage has a gain from the input terminal to its output terminal (base and collector, respectively, for this case). But the circuit gain, from the source to the output, takes into consideration the possible finite input resistance at the transistor input terminal. Due to the finite signal-source resistance, an attenuation results from the signal-source to the transistor input terminal. The example of this case of the common-emitter amplifier stage is considered here.

C.3.1. Gain from Base of Transistor to Output at Collector

The midband (frequency-independent) signal version of the circuit of Fig. C.1(b) is shown in Fig. C.4. It is obtained from the general circuit by converting dc voltages to zero. This includes the capacitor voltage, which ideally, remains at its constant dc (bias) value. The rule followed here is that if there is no incremental variation between any two nodes with a signal applied at the input, then the component between the two nodes is superfluous. In Fig. C.4, the output voltage is V_o = V_ce and the signal source is designated V_s along with its source resistance, R_s. In the case of an actual signal source, R_s is probably an equivalent rather than an actual resistor. Thus, it is assigned a lowercase subscript.

Neglecting the output resistance, the signal collector current for a signal voltage applied at the base reverts to (C.7), which is

with g_m = I_C/V_T [(C.12)]. I_C is the bias current with uppercase subscript, not the signal, with lowercase subscript. (Recall that signal voltage and currents can be, for example, periodic peak or RMS values or instantaneous values, as these are all proportional throughout the linear circuit. To be specific, as in the experiment on the common-emitter amplifier, we consider them to be periodic-signal peak values.)

The signal voltage developed at the collector is (still assuming that r_o >> R_C)

Equation C.16

From (C.16) and (C.7), the gain of the transistor in the circuit (input at the base of the transistor and output at the collector) is

Equation C.17

We note that the magnitude of the result is the dc voltage drop across the bias resistor divided by V_T. For example, for V_Rc = 5 V and V_T = 26 mV (room temperature), the gain magnitude is about 200. The BJT circuit is capable of providing very substantial voltage gains.

C.3.2. Overall Gain Magnitude from Signal Source Voltage to Output

The circuit input resistance looking into the base of the transistor, R_b, is the transistor input resistance, r_π (still neglecting r_b), in parallel with the bias resistor R_B, that is,

Equation C.18

The gain from the signal source to the output at the collector of the transistor is thus

Equation C.19

When the input-signal source resistance is large (R_s >> R_b), a good approximation for a_v is

Equation C.20

Further approximation can be made using R_B >> r_π, to obtain

Equation C.21

Finally, using β_ac = g_mr_π [(C.15)],

Equation C.22

This result is intuitive on the basis of I_b ≈ I_s, I_s ≈ V_s/R_s, and I_c = β_acI_b. The sequence of approximations for the gain magnitude is

Equation C.23

An additional approximation is with β_ac ≈ β_DC.

Note that the requirement for the voltage gain to be greater than unity is that R_s < β_acR_C. Thus, for sources with a large R_s, an amplifier design should have a high input resistance stage such as an emitter-follower stage. The emitter-follower stage is discussed in Unit C.9.

In the Project C1 the gain of the amplifier as a function of bias current, I_C, is measured using the circuit of Fig. C.1(a). This is made possible with the use of LabVIEW and the DAQ, with two output channels, to provide a signal source superimposed on the input bias voltage. In this case, the overall gain from the signal source is restricted (input bias and signal source resistor are the same) and a_v ≈ –β_acR_C/R_B. Since R_B of the circuit is roughly β_DCR_C, the gain is on the order of unity.