Current in CE and CB configurations of a transistor when output voltage is increased?
The following only applies if the BJT is in active mode and not saturated, obviously.

Let's say you measure \$I_{C_{CAL}}\$ and \$I_{B_{CAL}}\$ at the operating point \$V_{CE_{CAL}}=2\:\textrm{V}\$. If you hold \$I_B=I_{B_{CAL}}\$ you can predict the collector current as a function of \$V_{CE}\$ as:

$$I_C\left(V_{CE}\right)=I_{C_{CAL}}\cdot\frac{V_A+V_{CE}}{V_A+V_{CE_{CAL}}}$$

Conversely, if you hold \$I_C=I_{C_{CAL}}\$ then you can predict the base current as a function of \$V_{CE}\$ as:

$$I_B\left(V_{CE}\right)=I_{B_{CAL}}\cdot\frac{V_A+V_{CE_{CAL}}}{V_A+V_{CE}}$$



However, if you have circuit elements where neither \$I_B\$ nor \$I_C\$ is held constant, then you can still compute:

$$\beta\left(V_{CE}\right)=\frac{I_{C_{CAL}}}{I_{B_{CAL}}}\cdot\frac{V_A+V_{CE}}{V_A+V_{CE_{CAL}}}$$

And then use this computed value of \$\beta\$.



There are plenty of other factors that affect \$\beta\$. I'm sure you've heard of the "late effect", as well. There are effects at (1) low currents caused by carrier recombination effects at the surface and in the emitter-base space-charge layer and due to the formation of emitter-base surface channels; and (2) high currents where the minority carrier injection becomes significant and, because the space-charge must be neutral in the base region, the total majority carrier concentration must be similarly increased (by the same amount) and ultimately causing the collector current to asymptote. So there are other parameters to deal with these effects on \$\beta\$ at low and high collector current levels, which are not included in the above equations.



Note that I haven't mentioned \$V_{BE}\$ above.