Common-base current gain

The common-base current gain, written $\alpha$, is the ratio of collector current to emitter current in a BJT in active mode:

$\alpha =\frac{i_C}{i_E}$

Always less than 1 but very close to it, typically 0.98 to 0.995. Physically: in active mode almost every carrier the emitter injects survives the trip across the thin base and reaches the collector, so $i_C$ is nearly all of $i_E$, just slightly less because of the small recombination loss in the base (which is the base current).

$\alpha$ and the Common-emitter current gain $\beta =i_C/i_B$ describe the same device from two viewpoints, and the KCL constraint $i_E=i_C+i_B$ ties them together. Any one of $\alpha$, $\beta$, plus the KCL gives the other two. The conversions:

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

Derivation. From $i_E=i_C+i_B$ divide by $i_C$:

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

Taking the reciprocal gives $\alpha =\beta /(\beta +1)$. Solving that for $\beta$: $\alpha (\beta +1)=\beta \Rightarrow \alpha \beta +\alpha =\beta \Rightarrow \alpha =\beta (1-\alpha )\Rightarrow \beta =\alpha /(1-\alpha )$.

Sanity check: $\beta =100$ gives $\alpha =100/101\approx 0.990$; $\alpha =0.99$ gives $\beta =0.99/0.01=99$. Because $\alpha$ sits so near 1, a small uncertainty in $\alpha$ produces a huge swing in $\beta$. That extreme sensitivity is why designers avoid relying on the exact value of $\beta$ and instead use bias schemes that depend only on resistor ratios (Voltage-divider bias, Emitter degeneration).

$\alpha$ shows up directly in Common-base amplifier and Emitter follower gain expressions, where the relevant input node is the emitter, and in Emitter resistance via $r_e=\alpha /g_m\approx 1/g_m$.