Question

Define α and β. Derive the relation between then.

Answer 1

The dc common-base current ratio or current gain (α_dc) is defined as the ratio of the collector current to emitter current.

α_dc = \(\cfrac{I_C}{I_E}\)

The dc common-emitter current ratio or current gain (β_dc) is defined as the ratio of the collector current to base current.

β_dc = \(\cfrac{I_C}{I_B}\)

Since the emitter current I_E = I_B + I_C

\(\cfrac{I_E}{I_C}\) = \(\cfrac{I_B}{I_C}\) + 1

∴ \(\cfrac{1}{α_{dc}}\) = \(\cfrac{1}{​​{\beta_{dc}}}\) + 1

Therefore, the common-base current gain in terms of the common-emitter current gain is

α_{dc = \(\cfrac{\beta_{dc}}{1+{\beta_{dc}}}\)}

and the common-emitter current gain in terms of the common-base current gain is  β_{dc = 
\(\cfrac{\alpha_{dc}}{1-{\alpha_{dc}}}\)} 

For a transistor, α_dc is close to but always less than 1 (about 0.92 to 0.98) and β_dc ranges from 20 to 200 for most general purpose transistors.