C.2. Linear or Signal Model for the BJT

The primary function of a transistor in analog circuits is to produce a signal output current in response to a signal input voltage. In the case of the common-emitter circuits of Fig. C.1, the transistor input voltage is V_be and the responding output current is the collector current I_c. (Variable subscript conventions are covered in Unit 2. Uppercase symbols with lowercase subscripts denote RMS or peak magnitude of periodic signals.) The linear relation between the two variables is the transconductance, g_m. By definition, for the BJT

Equation C.7

In some simple amplifier circuits, this would be all that would have to be known about the transistor to perform a design or analysis. More generally, however, the model also includes input and output resistances, r_i = r_b + r_π and r_o, respectively. The transistor linear model, which includes these components and a load resistor (in this case, bias resistor, R_C), is shown in Fig. C.2. Applied input voltage V_be and responding I_c are indicated.

The input resistance relates the input signal base current I_b to the signal emitter – base voltage, V_be, that is

Equation C.8

Parameter r_b is the actual physical resistance through which the base current must flow to arrive at the true, internal physical base – emitter junction. The signal voltage across the internal base – emitter junction is . Parameter r_π is the linear relation between I_b and and is not a physical resistance.

We assume in the following discussion that r_π >> r_b. This is especially true in the low current range of our BJT transistor projects. As will be seen below, r_π is inversely proportional to bias collector current, I_C, whereas r_b is close to a constant and could be significant at currents corresponding to midrange or higher for the transistor. (As a rule, r_b must be taken into consideration when the model is applied in very high frequency applications.) Note that neglecting r_b compared to r_π is equivalent to , as assumed below.

The output-resistance parameter, r_o, accounts for the fact that total collector current, i_C, increases with increasing total collector – emitter voltage, v_CE. According to the output resistance parameter relationship, the current through this resistance is

Equation C.9

Including r_o, the current through the load, in this case bias resistor, R_C, is

Equation C.10

This current flows up through R_C such that V_ce is negative for positive V_be. Thus, the current associated with r_o subtracts from g_mV_be to reduce the current through R_C. For a positive V_be, there is an increase in the total v_BE, thus causing the total collector current to increase. The result is a decrease in the total v_CE and hence a negative incremental V_ce.

The two components of (C.10) are illustrated graphically in Fig. C.3. The output characteristics are for two values of base – emitter voltage: bias only, V_BE, and bias plus base – emitter signal voltage, V_BE + V_be. They are designated Bias and Signal. The solution to i_C and v_CE is constrained to the “load line,” which is i_Rc = i_C = V_CC/R_C – v_CE/R_C [(C.5)].

At a constant v_CE = V_CE, the change in the current-source current for the applied V_be is g_mV_be [(C.7)]. The net collector current change (signal current), I_c, though, is as given by (C.10); that is, it includes the component associated with r_o. Since the output characteristic slopes downward for decreasing v_CE, the actual transconductance decreases, but the linear model treats this effect with a constant g_m combined with the effect of the output resistance, r_o.

C.2.1. Determination of the Linear Model Parameters

We can relate the values of the two parameters in (C.10) to the SPICE model parameters using (B.7) with the substitution v_BC = –(v_CE – v_BE). This is

Equation C.11

Differentiating (C.11) with respect to v_BE (with v_CE = V_CE, V_ce = 0), g_m is found to be

Equation C.12

The approximate form only ignores a term on the order of I_C/V_AF, where V_AF >> V_T.

The output resistance is obtained with v_BE = V_BE (bias value) or V_be = 0. This is

Equation C.13

where, from (C.11), I_C(v_CE = V_BE) = I_Sexp(V_BE/V_T). For simplicity, the bias collector current, from (C.11), I_C = i_C(V_CE), is usually used for the calculation for r_o. Finally, a relation for r_π comes directly from (C.7), I_c = g_mV_be, and (B.41), I_c = β_acI_b. Equating the two gives

Equation C.14

Hence, from the definition r_π = V_be/I_b (with )

Equation C.15

Note that the right-hand side of (C.14) is the alternative current-dependent current source of the linear model of Fig. C.2.

As discussed in Unit B.9, β_ac can be slightly different from β_DC, but the distinction usually need not be made in analysis or design. This is due to the fact that β_DC tends to be quite variable among devices and that most analog designs are based on making the results as independent of β_DC as possible. Signal parameter β_ac will be used in the following, but it is understood that β_DC can be used in the calculations without serious penalty in precision.