Electrical – Why does the base-emitter voltage in NPN transistor become constant

The BJT collector current equation is

$$i_C = I_S\cdot e^{\frac{v_{BE}}{V_T}}\left(1 + \frac{v_{CB}}{V_A}\right)$$

where \$V_A\$ is the Early voltage. But, this formula is often written as

$$i_C = I_S\cdot e^{\frac{v_{BE}}{V_T}}\left(1 + \frac{v_{CE}}{V_A}\right)$$

Thus

$$\frac{\partial i_C}{\partial v_{CE}} = \frac{I_S\cdot e^{\frac{V_{BE}}{V_T}}}{V_A} = \frac{i_C}{V_A + v_{CE}}$$

This is clearly a non-linear function of the collector-emitter voltage and collector current so this cannot be interpreted as a conductance.

However, for small changes around some fixed value of collector current \$I_C\$ and collector-emitter voltage \$V_{CE}\$, we can write

$$I_C + i_c \approx I_C\left(1 + \frac{v_{ce}}{V_A + V_{CE}} \right) = I_C + \frac{v_{ce}}{r_o}$$

where

$$r_o = \frac{V_A + V_{CE}}{I_C}$$

We call \$r_o\$ the collector-emitter dynamic, or differential or small-signal resistance.

It is not a true resistance since it is not constant but, instead, varies with the operating point of the transistor as can be seen by the formula.

I thought the collector-base junction was reverse biased which means that the electric field created by the external bias adds to the potential barrier? How is it that holes spontaneously cross this huge potential barrier?

The built-in potential stops the diffusion current due to majority carriers diffusing from one side to the other. For example, electrons in the N material would diffuse to the P material (where they would recombine) if it weren't for the built-in potential. For the majority carriers, the built-potential is a barrier.

But in the N material of the base, holes are minority carriers and so, the built-in potential isn't a barrier at all. If a hole in the base exists long enough, it may be swept across the reverse biased base-collector junction by the electric field there and into the collector region.

When the base-emitter junction is 'on', lots of holes are injected into the thin and lightly doped base region so a large fraction of the injected holes exist long enough to 'slide down' the potential and into the collector region where they are majority carriers.