Below is an NPN transistor characteristics for Vbe versus Ic at different Vcb or Vce:

enter image description here

It seems like in the active region, Vbe vs Ic curves gets steeper with an increasing Vce.

The following equations relates Vbe to Ic in detail:

enter image description here

But from the above equations how can we conclude that the above curves become more steeper with increasing Vce?

The first part of the equation(Is) has many dimensional terms. Does any of them changes when the Vce increases? I want to relate the effect of Vce to the above curves through the above equations..

  • \$\begingroup\$ The Early Effect, caused by higher voltages across base-collector junction making that junction narrower and thus making the entire base region a bit narrower and thus more efficient at letting carriers move from emitter to collector without recombination, is not modeled in your equation \$\endgroup\$ – analogsystemsrf Oct 31 '17 at 15:14

The Early effect causes the effective base width to reduce as the voltage on the collector increases.

This in turn causes the collector current to increase for a given Vbe.

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