Current Amplification Factors: $\alpha$ and $\beta$

Common-base current amplification factor, $\alpha$. The common-base current amplification factor is defined as the ratio of the collector current to the emitter current:

Since, from Section 9.19, $I_C$ is always slightly LESS than $I_E$ (a small fraction of the emitter current is inevitably lost to recombination in the base, forming $I_B$), $\alpha$ is always slightly less than $1$ -- typically in the range $0.95$ to $0.99$ for a real transistor, the closer to $1$ the thinner and more lightly doped the base.

Common-emitter current amplification factor, $\beta$. The common-emitter current amplification factor -- by far the more commonly quoted and more practically useful of the two, since the common-emitter configuration (Sections 9.20, 9.22, 9.23) is the one most often used in practice -- is defined as the ratio of the collector current to the (much smaller) base current:

Because $I_B$ is only a small fraction of $I_E$, $\beta$ is a comparatively LARGE number for a real transistor, typically ranging from about $20$ up to several hundred -- $\beta$ is, in effect, a direct measure of how much a transistor amplifies base current into collector current, and is the single figure most often used to characterise a given transistor's current-gain capability.

Deriving the relation between $\alpha$ and $\beta$. Since $\alpha$ and $\beta$ are simply two different ratios of the SAME three currents, related by $I_E = I_B + I_C$ (Section 9.19), either can always be derived from the other. Starting from $\beta = I_C/I_B$, and substituting $I_B = I_E - I_C$ (from the current relation):

Dividing numerator and denominator by $I_E$, and using $\alpha = I_C/I_E$:

Rearranging this same relation the other way gives $\alpha$ in terms of $\beta$:

$\alpha = \frac{\beta}{1+\beta}$ …