An NPN BJT - from Spice to Ebers-Moll

Question

Wikipedia reports a simple model for a BJT in active mode (an approximation to the Ebers-Moll model):

\$I_E=-I_{ES}(e^{V_{BE}/V_T}-1)\$

(in the NPN case). Here, \$I_{ES}\$ is the Base-Emitter inverse saturation current. Finding it from a data sheet is difficult. But we have Spice models, and we can read parameters there. Here is a question about this same topic; an answer suggests using Spice parameters. However, three saturation currents exist: \$I_S\$ (transport saturation current), \$I_{SE}\$ (Base-Emitter leakage saturation current) and \$I_{SC}\$ (Base-Collector saturation current). Which one may be identified with the required \$I_{ES}\$ of the Ebers-Moll model?

Answer 1

See my discussion here for a brief overview of three equivalent DC (\$EM_1\$) models of the BJT. As you can see there, \$I_{ES}\$ is the term traditionally used in a current found in the injection version of the model.

In the injection version, the value of \$I_S\$ is also developed from reciprocity. It's called the "transistor saturation current" and is defined this way:

$$I_S = \alpha_F \cdot I_{ES} = \alpha_R\cdot I_{CS}$$

under the reciprocity relationship, which is both theoretically derivable and has also been experimentally observed (the seminal paper here being, B. L. Hart, "Direct Verification of the Ebers-Moll Reciprocity Condition," Int. J. Electronics, Vol. 31, pp. 293-295, 1971.)

The physical interpretation of \$I_S\$ is that it is the common portion of both \$I_{ES}\$ and \$I_{CS}\$. A PN junction saturation current consists of two different terms -- one each from analyzing the neutral regions. For a constant-doping, short-base diode, the saturation current is:

$$I_{SAT}=\frac{q\cdot A\cdot D_p\cdot p_{no}}{L_p}+\frac{q\cdot A\cdot D_n\cdot n_{po}}{W_B},~~~~\textrm{where}~ W_B \ll L_N$$

So \$\alpha_F \cdot I_{ES}\$ is the portion of the emitter-base saturation current that arises from analyzing the base region and \$\alpha_R \cdot I_{CS}\$ is the portion of the collector-base saturation current that also arises from analyzing the base region.

Nicely, this brings the need for four parameters in the injection model (at a given temp) down to just three. (I assume you know how to convert between \$\beta\$ and \$\alpha\$, of course.)

(I can't speak for the Wiki page and I'm not going to debate what it says there. I can only speak for what I know about.)