2.4. Transistor Output Resistance and Limiting Gain

The linear-equivalent circuit of Fig. 2.2 includes an idealization in that the output is a pure current source. In real transistors, the collector (BJT) or drain (MOSFET) current increases with increasing V_CE or V_DS. This is accounted for by including an output resistance, r_out, in the linear model, as added to the circuit in Fig. 2.5. For the BJT and MOSFET, respectively, the value of r_out is

Equation 2.9

Equation 2.10

where V_A is the characterizing transistor parameter. Note that this voltage dependence is not included in (2.1) and (2.2); these equations are consistent with the simplified circuit model of Fig. 2.2. Similarly, the voltage dependence will alter g_m from the simple forms of (2.4) and (2.5). This is discussed in Unit 4.

The actual gain, with a load R_L, which includes the output resistance, can be obtained from modification of (2.7) to include r_out in parallel with R_L as in Fig. 2.5. The result is

Equation 2.11

The parameter V_A of both transistors can typically be about 100 V. (In MOSFETs, the parameter is usually referred to as λ, which is the reciprocal, λ = 1/V_A.) A useful comparison between the devices is the maximum limiting gain of the common-emitter and common-source amplifier voltage gains, which applies for the case of R_L → ∞. The gain in this case is

Equation 2.12

Using (2.4), (2.5), (2.9), and (2.10), we obtain for the limiting gain:

Equation 2.13

Equation 2.14

Using sample numbers from above, the comparison gives a_{v lim}(BJT) ≈ –4000 and a_{v lim}(MOSFET) ≈ –400. The vacuum tube, type 6SN7, has a typical output resistance r_out ≡ r_p ≈ 7KΩ (p for plate), which leads to a limiting gain magnitude of about 21. (This is referred to as the μ of the tube.) You have to respect the amplifier designers of the vacuum-tube era when considering what was accomplished despite the limitations of these amplifying devices.

In modern integrated circuits, it is possible to implement load circuits, which have an effective R_L >> r_out such that the limiting gain can be achieved. This is particularly important in MOSFET amplifiers to make up for the relatively low value of g_m.