B.3. BJT Base, Emitter, and Collector Currents in the Active Mode

The discussion of the transistor mechanism of Unit B.1 is extended here to include base-current mechanisms, which exist in the real transistor, and the effect of base-width modulation as discussed in Unit B.2. Initially, the case of active-mode operation is discussed, and this is followed by the general-bias case, which includes the possibility of both pn junctions becoming forward biased.

In the device model for the bipolar junction transistor, the parameter β_DC is

Equation B.6

where I_B is the composite of all contributions to the base current. (The use of “dc” is consistent with SPICE.) This relates the output current, I_C, to the input current, I_B, for the transistor operated in the common-emitter terminal configuration.

In the following, the discussion is based on the npn transistor, as shown now with the schematic symbol in Fig. B.7. The npn is chosen over the pnp, as it is substantially more basic to BJTs than the pnp. This probably has to do mostly with the fact that a common-emitter stage is consistent with a positive power supply (a holdover from the vacuum-tube days), that the npn is superior in terms of frequency response, and that it has consistently been the device of BJT digital switches including the TTL. The pnp was used in the discussion above on the fundamentals of transistor action, as it is somewhat more intuitively satisfying to have the current and particle flow (where the illustration is with the hole) in the same direction.

The assignment of polarities of voltages V_BE and V_BC is consistent with forward bias for both the emitter – base and collector – base junctions. V_BC will be negative in active-mode operation. The assignment of V_CE is standard, as it will be positive in active-mode operation. Current directions are assigned to correspond to the actual directions of the currents in the forward-active mode.

Base currents do not couple between junctions. Rather, as suggested in Fig. B.8, a given base current is associated with a given pn junction. In the forward-active mode, a base-current component is added to the emitter current as indicated in Fig. B.7. However, since the base – collector junction is reverse biased, the base-current contribution to the collector current is negligible and (B.3) still represents the total collector current. The collector-current relation with the base-width-modulation effect (Unit B.2) now included becomes

Equation B.7

The equation has also been generalized to include the SPICE parameter n_F, the forward current emission coefficient (which is usually assumed to be 1).

The base current, for the forward-biased base – emitter junction, is made up of two terms; these are the ideal base current (because it has the same V_BE dependence as I_C) and the leakage base current. The leakage component is typically small enough to be neglected and often is, for simplicity, in practice.

In the absence of leakage base current, the base current is

Equation B.8

where I_SBE is the base – emitter saturation current. In the active mode, V_BE is positive, and typically, V_BE ≈ 0.6V. The exponential term is, for this case, exp(V_BE/n_FV_T) ≈ 10¹⁰ such that the –1 is quite negligible. The –1 is necessary for the current to go to zero for V_BE = 0. This current is due to (npn) injection and recombination of minority-carrier electrons in the base and injection and recombination of minority-carrier holes in the emitter. Base current could also be connected to injection of holes into the n-type emitter, which recombine at the emitter-contact metal – semiconductor interface. In any event, base – emitter junction base current is predominantly from injection into the emitter, as opposed to injection into the base.

For this case of no leakage current and for V_BC = 0, the forward-active current gain (or current ratio) is defined as β_DC ≡ β_F. Using (B.6), (B.7), and (B.8) (with –1 neglected), this is

Equation B.9

and I_SBE = I_S/β_F. Consequently, the ideal base current is normally written as

Equation B.10

Note that the base current does not have V_BC dependence as exhibited by the collector current. This is demonstrated to be valid in Project B.

The parameter β_F is the SPICE transistor-model β. Implicit in the assignment of IS (I_S) and BF (β_F) in the SPICE device model is the assignment of the ideal base-current saturation I_SBE = I_S/β_F. That is, there is no I_SBE (SPICE parameter) in the model. If the general active-region relation for I_C, (B.7), is now used in the β_DC definition, we obtain the β_DC relation for nonzero V_BC, namely,

Equation B.11

where β_F is a constant (SPICE BF) and β_DC = β_F for V_BC = 0.

In the real transistor, there is, in general, a component of leakage current. It is associated with recombination of holes and electrons in the depletion region of the base – emitter junction. In SPICE, the leakage-current component is characterized with parameters n_E (base – emitter leakage coefficient) and I_SE (base – emitter leakage saturation current). When added to the ideal component, the total base current is

Equation B.12

(These are both positive into the transistor for the npn.) The parameter n_E is ideally 2 (according to early, simplified theories), but in practice is typically about 1.5. When transistors are operated in or above the midrange of their rated current capacity, the leakage component becomes negligible.

For a small power transistor, the midrange starts at about 100 mA, such that the leakage term would be very significant at, for example, 1 mA. In the project on the transistor, which is designed to determine the SPICE parameters discussed here, a low-current range of less than about 1 mA is selected. The low current also avoids inadvertent rise in the temperature of the device during evaluation.

Similar equations that apply to the reverse-active mode can be obtained by direct substitution of the equivalent variables. The output current and input voltage, are, respectively, I_E and V_BC, while, for the reverse-active mode, V_BE = 0 or negative. Based on the equivalent substitution

Equation B.13

The minus sign comes from having assigned the current direction of the emitter current out of the transistor (Fig. B.7). The parameter I_S is common to (B.7) and (B.13). Equations (B.7) and (B.13) are combined to obtain the general case of V_BE and V_BC both positive (forward bias) in a following unit.

The reverse-operation base current is [the reverse-operation equivalent of (B.12)]

Equation B.14

where β_R is the dc current ratio (reverse-active mode) for the case of no leakage component of base current, and is given by β_R = I_E/I_B. In practice, the reverse ideal current gain is 0.1 < β_R < 10, and therefore, the reverse-operated transistor is not a useful configuration in, for example, analog amplifiers. However, the reverse-active mode equations are essential, as is shown below, for developing the general equations for the forward mode. The general form includes the possibility that the transistor will be out of the active mode.

Although V_AR (reverse Early voltage), I_SC (reverse leakage saturation current) and n_c (base – collector leakage emission coefficient) are officially SPICE parameters, we will not be concerned with these as they would rarely be a factor in the study of analog circuits.