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From my understanding,

The diffusion layer at a p-n junction is an equilibrium of two opposing forces on charge carriers due to 2 forces - Concentration Gradient and Electric Field caused by Bias and inturn by diffusion caused by Concentration. In steady-state (with or without bias) forces are perfectly balanced at an attained equilibrium (which is dynamic and changes with bias). Now the forces are all balanced, all that causes this dynamic resistance effect is the change in the size of the depletion layer thus resistance, not these forces since they've been balanced now.

Hopefully, that's the right picture of it. I'm trying to see how the depletion layer between Collector and Base changes with \$v_i\$ at Base (Common Emitter connection). I can see that depletion later EB is decided solely by the biasing 0 to \$v_i\$. What I cant explain is how this affects the depletion later BC.

What I'm trying to do is roughly get what happens to the resistance(\$\frac VI\$ rather than \$\frac {dV}{dI} \$) between Base and Collector, use \$I_e \approx I_c\$ and ohm's law to get graph between \$v_i\$ and \$v_o = v_C\$

Skip if needed: I searched up, all explanations have a "Current simply accelerates into Collector" which I clearly get from \$I_e \approx I_c\$ but don't get how that leads any relation between base current and collector current obtained or the output characteristics. I found this book in an internet archive by Millman & Halkias, read through chapter 5 where it reaches Output characteristics, didn't get it yet. Read other related questions like Why is Ib proporional to Ic in a bipolar transistor? , still don't get a qualititive picture other than the "Current is attracted by Forces" explaination or a "it gets into Semiconductor Physics and complex Math". The former explaination does not appeal to me because the forces are now balanced right? Isn't resistance by depletion layer causing this and not the forces? So what happens to this depletion layer?

Or more appropriately,

The Question: Can someone give a qualitative explanation of Output characteristics of a BJT using depletion layers instead of the regular "Current is attracted by Forces" one?

Thank you

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    \$\begingroup\$ One thing is the (ideally exactly proportional) Ic vs Ib relation, and another one is the Early effect (which is a statement on how Ic depends on Vce for a fixed Ib even in the linear zone where an ideal transistor would give a fixed Ic.) Are you having problems with both? \$\endgroup\$ Jan 10, 2021 at 19:12
  • \$\begingroup\$ @SredniVashtar , yes both. But I’m really only looking for an understanding of what happens with the depletion layer so I can continue studying from that lead. I was told in electronics.stackexchange.com/questions/541740/… that I can’t just take it to be 2 diodes and get \$I_c\$ vs \$v_o - v_i\$ \$\endgroup\$ Jan 10, 2021 at 19:14
  • \$\begingroup\$ Maybe seeing transistor action in terms of probabilities might help you? When you inject carriers at the emitter, the probability to be swept away and be collected by the collector is alfa. This leaves a small fraction of carriers that end up in the base. The bigger the base, the higher the probability to lose carriers inside it. The shorter the base, the higher will be Ic for a given Ie. Changing the width of the depletion layer can modulate the width of the base. Does this help? \$\endgroup\$ Jan 10, 2021 at 19:34

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Due to high doping ratios from base to emitter/collector, there exist 2 exponential curves at the same temp whose ratio gives the linear hFE ratio due to Conduction band bending.

Yet hFE increases >50% from very low currents then decreases typ. past the mid-range current.

Then into saturation, hFE decreases to a useable hFE=10% of max hFE when CB changes from a low leakage current sink to a conductive diode at Vce=Vce(sat) often rated at 10,20 and/or 50 if hFE>500.

The C-E conduction leakage may in a range of 100k to >1M for common types then into saturation the size of the chip rated in Watts is inversely related to the diode Rce quasi linear bulk resistance Rce<=1/Pmax rated down to 50% or so. Some as low as 10mOhm while PN2222 is 4 Ohms at Vce(sat) for 1/4W rating.

Some physics details.

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