B.5. Output Characteristics of BJT in the Common-Emitter Mode

The common-emitter terminal configuration can be considered the fundamental building block of BJT analog amplifier circuits. The common base and common collector are applied to some extent in more special-purpose roles, such as for high output resistance (common base) and low output resistance (common collector). Therefore, a study of the transistor in the common-emitter mode is basic to a study of analog circuits.

The output characteristic of the common-emitter transistor is defined as the output current (I_C) as a function of the output terminal voltage (V_CE) for I_B (or V_BE) held constant. It is very important to understand in the design of both analog and digital circuits. The output characteristic is experimentally explored in the project on parameter determination. The measurement circuit for obtaining the output characteristic is shown in Fig. B.12. This is also the circuit for measuring β_DC versus I_C, which is discussed below.

The equations presented in Unit B.2 are in terms of terminal voltages V_BE and V_BC. The common-emitter configuration has input voltage V_BE but output voltage V_CE. Therefore, it is preferable to eliminate V_BC in the equations in favor of V_CE, using V_BC = V_BE – V_CE.

When plotting the output characteristic, for example, from 0 < V_CE < 5 V, the range of V_BC is V_BE < V_BC < –(5 – V_BE). Since V_BE ≈ 0.5 V, V_BC covers the full range from strongly forward biased (out of the forward-active mode) to clearly reverse biased (forward-active mode). This requires that the I_C [(B.7)] and I_B [(B.12)] equations be modified to include all possibilities.

To obtain an equation for I_C under general biasing conditions, we start with the fundamental equation of transistor action of the BJT. This is

Equation B.20

This is a composite of (B.7) and (B.13) in which the base-modulation effects are neglected. Both of these equations are for I_C and I_E in the ideal transistor. (In this unit and beyond, we assume that n_F = 1.) Sometimes referred to as the linking-current equation, this is symmetrical in I_C and I_E and neglects all components of base current and dependence of the base width on V_BC and V_BE.

To add an additional degree of applicability to the real transistor, base-width modulation must be added. We are interested in a relation between currents and voltages for the transistor in the forward-active mode. In this case, V_BE is fixed at 0.4 < V_BE < 0.6 V. On this basis, base-width modulation of the emitter – base junction can be neglected. On the other hand, the relation must apply for a V_BC range, which includes relatively large negative values. Therefore, the effect of base-width modulation of the collector – base junction must be included in a modification of (B.20). If base-width modulation due to the dependence of base width on V_BC is added and the substitution V_BC = V_BE – V_CE is now made, (B.20) becomes:

Equation B.21

The general equation must include any significant base current that contributes to collector current, that is, for when the base – collector junction becomes forward biased. For this we use (B.14) with base leakage current neglected. (Leakage current is neglected throughout this unit.) This is

Equation B.22

The –1 term has been retained in order for the equation to apply at all possible polarities and values for V_BC = V_BE – V_CE, including zero where this contribution of collector current is zero.

We note that this base current is in a loop (Fig. B.8) through the base – collector junction and is positive out of the collector terminal; I_C of (B.21) is positive into the collector terminal as in Fig. B.7. Thus, the component of base current from (B.22) subtracts from the collector current to give

Equation B.23

Again, for simplicity, this equation neglects the leakage current associated with the collector – base junction [I_SC term (B.14)], which is only a fair approximation for the level of I_C at which we will obtain an output characteristic in the parameter extraction project on the BJT.

A plot of this equation for I_C as a function of V_CE can be obtained for constant V_BE or constant I_B. For the latter, V_BE is also a variable such that the plot requires, in addition, a solution for V_BE as a function of V_CE for constant I_B. In the project on the output characteristic, the measured data and a SPICE solution are compared during the measurement. The input circuit provides a more or less constant I_B. However, the SPICE solution, which is computed by LabVIEW, is exact. It allows for a variable I_B and associated variable V_BE. The SPICE solution formulation used by LabVIEW is outlined below in Unit B.7. The SPICE solution is also explored in the project Mathcad file.

After measuring the output characteristic, the segment of the resulting data array from the active region (V_CE > V_BE) is extracted. From these data, a straight-line curve fit is obtained by LabVIEW, which produces a number for the slope. From (B.23), we obtain an equation for the active region from letting V_CE > V_BE, with the result

Equation B.24

The slope for the active-region equation is slope = dI_Cact/dV_CE = I_C/V_AF. Thus, V_AF can be calculated from V_AF = I_C/slope, where I_C is the value taken from the data array for V_CE = V_BE.

For the special case of V_CE = V_BE, one can obtain a value for I_S from the relation

Equation B.25

After V_AF and I_S have been obtained, an iteration on β_R will produce a final curve fit to the output characteristic, thereby providing a number for β_R. In this manner, all three parameters of this unit are obtained: I_S, β_R, and V_AF.

An example of a SPICE output characteristic (calculated in Mathcad), which uses the measured transistor-model parameters, is shown in Fig. B.13. Also shown is the calculated active-region plot.

The low-voltage region where I_C < I_Csat is called the saturation region. The voltage and current in this region are V_CEsat and I_Csat. In the derivation of the general equation for I_C, (B.23), the leakage current of the base – collector junction was neglected. If this is not valid, the reverse β_R, which is determined through curve fitting, is somewhat artificial. For example, suppose that the leakage current totally dominates the collector – base current. For this case, the general I_C equation is

Equation B.26

where a new definition of a sort of β_R is defined in the relation I_SC = I_S/β_Rleak. A curve fit to the measured output characteristic would give β_Rleak. However, because n_C > 1, the result depends on the level of collector current at which the measurement is made. If, in a given measurement, the β_Rleak is interpreted as β_R, it would function as β_R in SPICE (when assigning BR to the model), but it would only be valid in a simulation for collector currents close to that of the measurement.