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In this plot, when \$V_{ce}\$ increases in the active region, shouldn't \$I_c\$ also increase as more electrons are getting diffused into the collector?

Then, why is it that for a large range of \$V_{ce}\$ , \$I_c\$ doesn't seem to change at all?

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  • \$\begingroup\$ Because the base collector depletion region widens with more collector voltage. \$\endgroup\$ – Andy aka Dec 15 '18 at 20:47
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I'm just an undergraduate student, so i'm not sure my answer is right. Your plot is showing Ic-Vce curve in common emitter BJT when Vbe is constant. So you can think Vcb changes by the amount of the change of Vce.
If you consider energy band diagram in active region, B-C junction will show reverse bias. In reverse bias, majority carrier(holes in P or electrons in N) wander into depletion region and will be swept into the other side by electric field(drift current). And this process is not affected by the magnitude of Vce. This is because in respect of majority carriers do not see any energy barrier in reverse bias(active mode). Therefore in ideal situation , the plot would be flat. The slight increase is due to base width modulation(Early effect), and if Vce further increase Ic would go up very high(Breakdown or Punch-through occurs)

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