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Regarding the reverse saturation current \$I_{ES}\$ aka \$ I_{EO}\$ of an NPN transistor, according to Ebers–Moll Model:

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

Approximated form becomes:

\$I_E \approx I_{ES} (e^{V_{BE}/V_T} -1)\$ or even more simply:

\$I_E \approx I_{ES} (e^{V_{BE}/V_T})\$

Let's say in a design somehow I want to quantify \$V_{BE}\$ drop more accurately for a desired 1mA emitter current, rather than just taking \$V_{BE}\$ as 700mV.

And in that case knowing the above formula and \$I_{E}\$ = 1mA, \$V_{T}\$ = 26mV I want to quantify \$V_{BE}\$ as:

\$V_{BE} \approx 26×10^{-3}×ln(10^{-3}/I_{ES})\$

So above to quantify \$V_{BE}\$, I'm missing the value for the parameter \$I_{ES}\$.

Here is the datasheet for 2N2222 transistor.

Isn't this parameter called reverse saturation current between the base-emitter junction? Either I cannot find this parameter in the data-sheet, or it is just not there. Is there another name for \$I_{ES}\$ or it is not presented?

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It's not presented.

You might be able to extract it by curve fitting Figure 4, which shows the typical relationship between \$V_{BE}\$ and \$I_C\$.

Even easier, you can find a SPICE model for 2n2222, which will likely include a reasonable value of \$I_{ES}\$ determined by a similar curve-fitting technique.

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