# How to include the Early effect in NPN BJT current equation?

To understand the NPN current equation derivation, I have been studying my lecture handout and the emitter current equation accordingly is found as :

$$$$I_E= q n_i^2 A(\frac{D_p}{N_{dE}W_E}+\frac{D_n}{N_{aB}W_B})(e^{\frac{qV_{BE}}{KT}}-1)$$$$

The collector current equation: $$$$I_C= q n_i^2 A(\frac{D_n}{N_{aB}W_B})(e^{\frac{qV_{BE}}{KT}}-1)$$$$

This already includes the effect of increasing the collector-emitter voltage in the factor: $$$$\frac{1}{W_B}$$$$

As in, if we increase the reverse bias voltage the junction width increases (Wb decreases) which further increases the slope of concentration gradient of electrons in the base region and this also increases diffusion current.

However , in the lecture the professor changes the $$$$I_C= q n_i^2 A(\frac{D_n}{N_{aB}W_B})(e^{\frac{qV_{BE}}{KT}}-1)$$$$ to $$$$I_C= I_s e^{\frac{qV_{BE}}{KT}}(1+\frac{V_{CE}}{V_A})$$$$ to include the Early effect or the increase in current due to changing collector-emitter voltage.

$$$$I_s= q n_i^2 A(\frac{D_n}{N_{aB}W_B})$$$$ and -1 removed due to a higher operating point.

But why is this factor $$$$(1+\frac{V_{CE}}{V_A})$$$$ necessary in the current equation?

Doesn't $$$$I_s= q n_i^2 A(\frac{D_n}{N_{aB}W_B})$$$$ already cover that ?

• I see you are trying to make sense of your lecture. And it is hard to argue, through you, to your lecturer. So there is no possible way to "come to terms" unless everything said makes absolute sense in all circumstances. And they don't. I hope you will look over Gummel & Poon's 1970 paper. It unifies the high current regime, the Early Effect, and the forward transit time vs collector current into a single concept. Meanwhile, I can just suggest that the factor should be $1+\frac{V_{_\text{BC}}}{V_{_\text{A}}}$ and not as you show.
– jonk
Jan 12, 2022 at 8:42
• I am unfortunately a bit confused by the notations used in the paper "Gummel and Poon's 1970 paper" especially in eber's moll model equations (for ex the factor of 1/(1-alpha_n*alpha_i) and usage of charge in further equations , is there any easy way to understand this ? Jan 19, 2022 at 16:46