Relation Between $\alpha$ and $\beta$

The two current-gain figures defined for the two configurations -- $\alpha=I_C/I_E$ in common base, $\beta=\Delta I_C/\Delta I_B$ in common emitter -- are not independent; they are related by $\alpha=\beta/(1+\beta)$, equivalently $\beta=\alpha/(1-\alpha)$. Because $\alpha$ is always very close to 1 (typically 0.95-0.99), this relation is why $\beta$ comes out so much larger than $\alpha$ -- a small change in $\alpha$ near 1 corresponds to a large change in $\beta$, which is exactly why $\beta$ is the far more s …