B.6. SPICE Solution for I_C versus V_CE of the Measurement Circuit

All components of base current are those from (B.12) (forward mode) and (B.14) (reverse mode). These are summed together to obtain the following expression for the general bias case:

Equation B.27

In the measurement circuit of the project on the output characteristic of the BJT, the circuit will provide a current source at the input and I_B (to a good approximation) will be a constant. However, in the event that a precision solution for I_B and V_BE is required, it can be obtained by equating (B.27) to I_B from the input circuit equation (Fig. B.12), which is

Equation B.28

In the Mathcad project file, a solution for V_BE as a function of V_CE is obtained (using a root finder) for a given V_BB and R_B. The result for V_BE (at a given V_CE) is used in (B.23) for a solution for I_C, and hence the output characteristic is obtained. For this, the leakage components of base current are neglected.

In the LabVIEW output characteristic measurement project, a SPICE solution is obtained along with the measurement. The solution is obtained with LabVIEW using an iterative solution. The equations are (B.28) and (B.27) (without the leakage terms) solved for V_BE. This is

Equation B.29

The iterative method consists of guessing an initial V_BE and solving for I_B from (B.28). This is used in (B.29), with the given V_CE, to obtain a better value for V_BE. The new V_BE is then put back into the circuit equation, (B.28), and so on, until V_BE stops changing significantly. This V_BE is then used in (B.23) for a solution for I_C at the specified V_CE. For the complete output characteristic, the solution is repeated in increments of V_CE for a range from zero up to a specified maximum. This is accomplished in LabVIEW with a Formula Node in a While Loop. The analytical formulation is explored in the project Mathcad file.

From (B.29), we note that over the full range of 0 < V_CE < 5 V, for example, the limits of V_BE are for V_CE = 0,

Equation B.30

while for large V_CE,

Equation B.31

The case of (B.30) uses β_F >> β_R. For example, for β_F/β_R = 100, the difference V_BE(hi) – V_BE(lo) ≈ V_Tln(β_F/β_R), which is about 120 mV. In the base circuit equation (B.28), this change would be minor compared, for example, with a base bias voltage, V_BB, of 5 to 10 V. Thus, base current is close to a constant over the full range of V_CE, based on Chan1_out ≡ V_BB = 5 V or greater, as dictated by the design goal. That is, V_BB is much greater than the change in V_BE as V_CE moves from (B.30) to (B.31).

We note that, consistent with the limit of (B.31), the solution for I_B is from

Equation B.32

Therefore, I_B and V_BE are constant and I_C only varies with V_CE due to the V_CE dependence in (B.24). The value of the limit V_CE is quantified in the next unit.