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