I am currently attempting to measure the parameters of a "vintage" germanium BJT in the hope of creating an Ebers-Moll model from the data. So far I have been unable to find a comprehensive source on the topic as it is slightly out of the region of my expertise.

So far my primary source of advice has come from "and yet another Definitive Handbook of Transistor Modeling" which is fantastic, but has no author, no date, and some of the pages are shuffled.

In search of a more up to date source I have been using the doctoral thesis of Martin Linder, "DC Parameter Extraction and Modeling of Bipolar Transistors". This goes too in depth in some areas and does not cover the basics as they are probably not worthy of writing in a thesis.

This did however lead me to the original paper by Gummel and Poon, "An integral charge control model of bipolar transistors" (ran out of links). This has lots of relevant information but is written like a research paper so not very useful when trying to practically apply it.

Has this information been covered in a text book anywhere? A rigorous method of extracting \$I_S,\; \beta_F,\; \beta_R,\; N_F,\; N_R\$?

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    \$\begingroup\$ nxp.com/wcm_documents/models/bipolar-models/mextram/… That's one of the most sophisticated BJT models, by the way. \$\endgroup\$ Commented Nov 23, 2015 at 16:39
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    \$\begingroup\$ Also for GP: ftp.elo.utfsm.cl/~lsb/elo102/ejercicios/GP_DOCU.pdf People who write such detailed guides don't extract E-M, because it's not used. GP is used in SPICE. And they write the guide assuming you're having some equipment like curve tracers and you can run IC-CAP. If you want to do it the old-fashioned way... you're going to bite the bullet and read old fashioned papers. \$\endgroup\$ Commented Nov 23, 2015 at 17:11
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    \$\begingroup\$ Since GP reduces to EM you could extract just what you care about, but I don't think I can find any guide [written in the last 30 years] that does't involve equipment you probably don't have (or you wouldn't be asking this question). Maybe some graduate textbook on semiconductor fabrication would cover the theory of the parameter extraction at a basic/textbook level. I don't see undergraduate textbooks bothering with this because you buy stuff with datasheet at that level. \$\endgroup\$ Commented Nov 23, 2015 at 17:23
  • \$\begingroup\$ This book has chapter on it, but it's somewhat focused on RF. It's wort reading at least that intro page though. \$\endgroup\$ Commented Nov 23, 2015 at 17:29
  • \$\begingroup\$ Thanks for all the links @RespawnedFluff that's exactly what I was looking for. I'm using a DAQ from to measure \$V_{BE}\$ and \$V_{CE}\$ with known resistances between those voltages and the sources to calculate the currents. The resolution is not perfect but it's enough to get a good approximation. \$\endgroup\$
    – loudnoises
    Commented Nov 24, 2015 at 11:59

2 Answers 2


Ebers-Moll Model

$$ \begin{align} I_{\mathrm{b}} & = \frac{I_{\mathrm{s}}}{\beta_{\mathrm{f}}}\left(\mathrm{e}^{\frac{V_{\mathrm{eb}}}{NV_{\mathrm{t}}}} - 1\right) + \frac{I_{\mathrm{s}}}{\beta_{\mathrm{r}}}\left(\mathrm{e}^{\frac{V_{\mathrm{eb}} - V_{\mathrm{ec}}}{NV_{\mathrm{t}}}} - 1\right) \\[0.9em] I_{\mathrm{c}} &= I_{\mathrm{s}}\left(\mathrm{e}^{\frac{V_{\mathrm{eb}}}{NV_{\mathrm{t}}}} - 1\right) - I_{\mathrm{s}}\frac{\beta_{\mathrm{r}} + 1}{\beta_{\mathrm{r}}}\left(\mathrm{e}^{\frac{V_{\mathrm{eb}} - V_{\mathrm{ec}}}{NV_{\mathrm{t}}}} - 1\right). \end{align} $$

Model parameters:

  • Saturation current \$I_\mathrm{s}\$
  • Ideality factor \$N\$
  • Thermal voltage \$V_\mathrm{t}\$
  • Forward gain \$\beta_\mathrm{f}\$
  • Reverse gain \$\beta_\mathrm{r}\$

Direct parameter extraction

The most straight-forward way of finding parameter values for the Ebers-Moll model from measurements is using direct extraction.

The below figure illustrates a Forward Gummel measurement of the 2N3906 BJT, which is when \$V_\mathrm{ec}\$ is kept a constant potential and \$V_\mathrm{eb}\$ is swept over a range. For this figure \$V_\mathrm{ec} = 0.3\ \text{V}\$. (As an aside I chose this badly as you want the value of \$V_\mathrm{eb} - V_\mathrm{ec}\$ to be small for the Ebers-Moll model to be a good approximation).

Ebers-Moll model parameter estimation

In the middle of the measurement we see an ideal region which appears linear but is actually exponential as the current is on a logarithmic axis. The Ebers-Moll model only captures the exponential behaviour of the BJT and not the high and low-current regions, so the parts at the top and bottom can be ignored. From the middle of the measurement 3 out of 4 parameters can be determined.

The gradient of the ideal region controlled by the parameter combination \$ N V_\mathrm{t} \$. Inspecting the Ebers-Moll equations when biased in the forward-active region (i.e. \$V_\mathrm{eb} - V_\mathrm{ec}\$ is negative) we can see $$ I_\mathrm{c} = I_\mathrm{s} \mathrm{e}^\frac{V_\mathrm{eb}}{NV_\mathrm{t}} $$ where \$NV_\mathrm{t}\$ is the gradient of the exponential current with respect to \$V_\mathrm{eb}\$. The thermal voltage can be found by measuring the junction temperature during the current measurements and applying this formula.

Also from this equation we can see that when the exponent is 0, \$I_\mathrm{c} = I_\mathrm{s}\$. When \$V_\mathrm{eb}\$ approaches 0 however the BJT does not behave ideally, so to find \$I_\mathrm{s}\$ we extrapolate from the ideal region.

Finally the gain can be found from the approximation \$I_\mathrm{c} = I_\mathrm{b}\beta_\mathrm{f}\$, i.e. by selecting a point in the ideal region and finding the multiplicative difference.

For reverse measurements \$V_\mathrm{eb}\$ is simply made negative which reveals a similar plot but with the reverse gain factor \$\beta_\mathrm{r}\$.


Ian Getreu, 'Modeling the Bipolar Transistor', 1976, Tektronix


The main problem you will find when trying to get the parameters of a BJT is that they are quite dependent on many variable conditions, as temperature. However, there is no problem to get those values in a given time.

The simplest way to get forward beta is to provide a current smaller than the saturation one in active mode, so measuring the current in base and emitter, you can directly calculate beta:

beta =Iemitter/Ibase - 1

The saturation current is obtained just in the point where the result of the equation isn´t constant anymore and starts to decrease.

I have never used BJT in reverse mode, but those parameters will probably be calculated in an analog way.

  • \$\begingroup\$ I have been attempting to use a DS18B20 to approximate the temperature, which I know fails as it is only the external temperature, but I am hoping it puts it into the right area. The other methods I have been using are extracting gradients from \$V_{CE}\$ vs. \$I_C\$ and \$V_{BE}\$ vs. \$I_B\$. I was just hoping there was a textbook that might cover the area. \$\endgroup\$
    – loudnoises
    Commented Nov 23, 2015 at 16:26
  • \$\begingroup\$ The book Microelectronic Circuits, from Sedra and Smith, has a whole chapter dedicated to BJTs and thereis explained the Ebers-Moll model \$\endgroup\$
    – Zero point
    Commented Nov 23, 2015 at 16:50

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