What are the limits of the Ebers-Moll model for large currents?

Question

I learned that the Ebers-Moll equation Ic = Is(T) ∗ exp(Vbe/Vt) "predicts" the collector current to rise exponentially with the base-emitter Voltage.

Experimenting with LTSpice for a LED current sink I apparently ran into the limits of the Ebers-Moll model / equation:

Instead of a straight line into ... lets say to 1000 A, there is a knee at around only 100 mA to 1 A.

I didn't expect to hit the limits of Ebers-Moll at such "low currents". Assuming the simulation results are correct.

This doesn't look like the early effect either. I don't know what's happening here.

Switching to a "more powerful" transistor doesn't show different results, nor does increasing R1 to 0.1 Ohms.

1. Did I run into the limits of the Ebers-Moll equation at around 100 mA? Or did I make a mistake?

2. Could somebody please explain what limit I ran into here? Why is the current not increasing exponentially anymore?

3. What model explains this effect?

Thanks!

Answer 1

In general, DC BJT models include a lot more than some simple Shockley diode model that you show.

One factor that I discuss here is the knee that occurs due to the increasing importance of minority carriers and their impact on majority carriers. The knee is modeled by the setting of IKF. I also mention there the significance of the collector resistance, which is modeled by the setting of RC.

However, the IKF factor (knee current due to current crowding) is only going to change the slope by a factor of 2, which on a log10 scale over a range of many magnitudes won't be very visible. So it's not the key factor.

However, the value of RC is \$85.6\:\text{m}\Omega\$ and the value of RB is \$810\:\text{m}\Omega\$. Both of which seem likely candidates for a problem at high currents. Also, while RE is less than \$1\:\text{m}\Omega\$, it's still a possible factor to worry over.

So let's just set all of them to near-zero and adjust IKF from the default value of \$1\:\text{A}\$ to 10X and 100X to see what happens:

Looks quite linear on the log-scale, now! The only difference is where the knee takes place on the three curves. The lower green one (the default case) bends over at \$1\:\text{A}\$. The middle blue one at \$10\:\text{A}\$. And the upper red one at \$100\:\text{A}\$. (This is the factor of 2 point of change.)

(The bottom-most curve is yours, for comparison.)

So you need to take into account bulk resistance for the pins. And you may also want to take into account the knee current.