# Is there a derivation of the Early effect?

I can't seem to find a derivation of the following equation:

$$I_C=I_Se^{\frac{V_{BE}}{V_{T}}}\left(1+\frac{V_{CE}}{V_A}\right)$$

Is there a way to derive this equation?

Edit: The equation describes a bjt transistor, so I have included a bjt. Also a plot of Ic against Vce to demonstrate the early effect.

• If you are going to ask where equations come from, it is helpful to draw a circuit that the equation describes. Although some people will see some of the terms (such as Vbe, Vce) and be able to deduce what the equation is describing, some will just see it as useful as if you replaced everything with a, b, c, d etc. Please add some context to the question.
– MCG
Oct 16, 2019 at 8:02
• @MCG I have included the relevant circuit and a plot to illustrate the early effect. Oct 16, 2019 at 8:12
• Are you OK with the derivation of the ebers moll equation and the simplification used in your equation? Oct 16, 2019 at 9:17
• @Andyaka yes I understand Oct 16, 2019 at 9:22
• @dilinex "J. M. Early, "Effects of Space-Charge Layer widening in Junction Transistors," Proceedings IRE, Vol. 40, pp 1401-1406, November, 1952. J. M. McCalla, "Computer-Aided Design of Integrated Bandpass Amplifiers," university of California, Berkeley, Ph.D Disertation, June 1972. "Transition Region Capacitance of DIffused pn Juntions" IEEE trans. Electron Devices, voll ED-20, pp 290-298, March 1971. "P. E. Gray, et al, "Physical Electronics And Circuit Models of Transistors," SEEC Vol 2., Section 9.1, J. Wiley 1964. And A B Philips, "Transistor Enjineering, "Section 9.2, McGraw_Hill, 1962.
– jonk
Oct 16, 2019 at 9:48

If $$\A\$$ is the angle at A, and $$\I_{CA}\$$ is the current at $$\V_{CE}=0\$$ which approximately equals $$\I_{C}\$$ (with no Early effect), then
$$\\tan(A) = I_{CA}/V_A\$$
If $$\\Delta I_{CA}\$$ is the increase in the value of $$\I_c\$$ due to the Early effect then also,
$$\\tan(A) = (I_{CA} + \Delta I_C ) / (V_A+V_{CE})\$$
Now solve for $$\(I_{CA}+ \Delta I_C)\$$ you will get the above equation.