# An NPN BJT - from Spice to Ebers-Moll

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?

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.)

• Very interesting. I just ordered the book you quoted in your other post. Honestly, for now I would content myself by finding the $I_{ES}$ parameter from the Spice model. There has been an answer which said it to be just $I_{SE}$ from Spice, and I tried that with an MPSA18, but with no satisfactory result - at 25 °C and $V_{BE}$ = 0.6 V, I find $I_E$ about 1.9 mA, while the data sheet quotes just 1 mA. Spice says $I_{SE}$ = 166.7 fA. (Model from ON Semiconductors.) The answer has later been deleted... Am I doing something wrong or is it just a matter of approximation? – Enrico Dec 12 '16 at 21:37
• @Enrico The $I_S$ used in Spice is taken from the transport model (equivalent to injection.) I've had very little problem getting exactly expected values on paper vs LTSpice, for example. So you'd need to provide a completely worked example problem to illustrate the details for me before I could help you find your way. Datasheets may be at some variance with the Spice models, though, since there are different goals there (max, min, typical) and a model only expresses a single instance. There are web pages on how to use a datasheet to get spice parameters, but really you need part testing. – jonk Dec 12 '16 at 22:12