Common-emitter current gain

The common-emitter current gain, written $\beta$ (and called $h_{FE}$ on datasheets), is the ratio of collector current to base current in a BJT operating in active mode:

$\beta =\frac{i_C}{i_B}$

It is the headline parameter of a BJT. It says how much collector current you get for a given base current — the device’s current amplification. Typical values are 50 to 100, with some processes reaching the high hundreds or even ~1000.

Where it comes from

In active mode the emitter injects a large carrier current into the thin base (Minority-carrier injection). Almost all of those carriers diffuse across and become collector current; a small fraction recombine in the base on the way and that lost fraction is the base current. So $i_B$ is small, $i_C$ is large, and their ratio $\beta$ is large. A bigger $\beta$ means a thinner, more lightly-doped base — fewer carriers lost to recombination.

A small fraction of the emitter current is lost to recombination in the base, becoming the base current; the rest reaches the collector.

Relation to α and KCL

A BJT has only three terminals, so by KCL the emitter current is the sum of the other two:

$i_E=i_C+i_B$

The Common-base current gain $\alpha =i_C/i_E$ is the other ratio. The two are not independent — knowing one (plus the KCL constraint) gives the other. Starting from $i_E=i_C+i_B$ and dividing through by $i_C$:

$\frac{i_E}{i_C}=1+\frac{i_B}{i_C}=1+\frac{1}{\beta }=\frac{\beta +1}{\beta }$

so $\alpha =i_C/i_E=\frac{\beta }{\beta +1}$, and inverting that,

$\beta =\frac{\alpha }{1-\alpha }$

A worked check: $\beta =100\Rightarrow \alpha =100/101=0.990$. Conversely $\alpha =0.99\Rightarrow \beta =0.99/0.01=99$. Because $\alpha$ is so close to 1, a tiny change in $\alpha$ swings $\beta$ enormously — which is also why $\beta$ varies widely between devices and with temperature, and why good bias circuits (Voltage-divider bias, Emitter degeneration) are designed to be insensitive to $\beta$.

From $\beta =i_C/i_B$ you also get $i_E=(\beta +1)i_B$, the form used throughout BJT DC analysis. The factor $(\beta +1)$ between base and emitter currents is exactly what powers the Resistance reflection rule.