# How to prove the Early effect in BJTs?

I am searching for some sort of proof leading up to Early effect correction in the BJT current formula and the need to include a factor of current $$1+\frac{V_{CE}}{V_A}$$.

I have tried reading "Derivation and Interpretation of a Generalized Charge-Control Theory and Reciprocity for a Bipolar Transistor" but sometimes concepts are too heavy for me to understand. I am looking for an easily understandable but complete explanation.

• Have you yet bothered to read J. M. Early's "Effects of Space-Charge Layer Widening in Junction Transistors?" It covers how the collector-barrier thickness affects collector voltage and its impact on the output resistance as well as the feedback voltage, as well as showing that the earlier base resistance (combining two effects) at low frequency fails at high frequency. It's a good paper. You say you want "complete" and I'd like to suggest that saying so limits the "easily understandable" part. You can simplify only so far. Beyond some limit, it becomes unusably distorted.
– jonk
Commented Feb 1, 2022 at 21:04
• Also, just another thought came to mind. Basewidth modulation is generally assumed (incorrectly) to be when operating in a linear mode and where constant doping holds. These are not very accurate assumptions but they are justified because basewidth modulation is a 2nd order effect. A first-order analysis is acceptable, given that. So it's already no longer "complete" because the justifications hang already upon assumed simplifications we know are not true and not accurate.
– jonk
Commented Feb 1, 2022 at 21:13
• I have tried to read it but I couldn't get past the first equations that were specified in matrix form for $$I_C , I_E$$ and even after it became more complex..is there anything I can read before it to make sense out of it ?I have been reading a relatively simple derivation for ebers moll model where you measure charge density as a function of x(distance) and derive the current from it by subsequently adding holes and electrons part. Commented Feb 1, 2022 at 21:13
• Which page are you looking at, in particular? And what microelectronics books are you reading from? Also, have you attempted your own Taylor's series expansion of the basewidth based around $V_{_\text{BC}}=0$? (We are starting to have a discussion and that's frowned on here. In any case, I have other work I need to do, today. Also, as I may be able to poke a thought or two here and there, it remains that I'm also limited in my own perspectives. I'd need to re-compose my thoughts and hear more from you before I'd attempt any kind of answer. What you've written so far tells me too little.)
– jonk
Commented Feb 1, 2022 at 21:20
• (The Early paper spans from page 1401 to 1406 in the Proceedings of the I.R.E. in the November 1952 issue.)
– jonk
Commented Feb 1, 2022 at 21:22

You must clearly distinguish between

• the physical explanation of the Early effect (base width modulation), and

• the artificial visual (graphical) method to express this effect using the definition of the Early voltage VA. This voltage VA is an artificial quantity which cannot be verified by direct measurements. It is nothing more than a graphical method to mathematically capture the influence of the slope of the Ic=f(Vce) characteristic curves in a simple way.

... and the need to include a factor of current 1+VCE/VA

The term is included in the equation of betaF.

Here is a sample sheet for the simulation of the Early Effect. (NB: formulas are complicated)

Here is the case where Va (Early effect) is equal to 1 V (example).

The factor (1+Vcc/Va) has been included.

The formula for Ic has been included here for completeness ...
We can seen that formulation is "very" complicated, although the problem is "simple".

Here is the picture with Va = 1 V shown.

Here is the picture with Va = 5 V shown.