Derive a relation between current gain of common emitter amplifier.

Explanation

Current gain `alpha`is defined as the ratio of collector current to the emitter current at constant collector voltage.
Current gain `beta` is defined as the ratio of the collector current to the base current at constant collector voltage.
i.e. `beta=(I_c/I_b)_(E_(ie))`
Relation between `alpha and beta`.
For n-p-n and p-n-p transistor, we have
`I_e = I_b + I_c`
`or I_e/I_c = I_e/I_c + 1`...(i)
Since current gain of common emitter amplifier,
`alpha = I_c/I_e`
And current gain of common base amplifier,
`beta = I_c/I_b`
So Eq.(i) becomes
`1/alpha = 1/beta + 1`
`or 1/beta = 1/alpha - 1`
`or beta = alpha/(1-alpha)`