Question

Obtain the relation between the current amplification factor $\alpha$ and $\beta$ of a transistor

Answer 1

The relation between the current amplification factors $\alpha$ and $\beta$ of a transistor.
The current amplification factor or current gain of a transistor is the ratio of output current to the input current. If the transistor is connected on base mode the current gain $\alpha =\frac{I_C}{I_E}$ and if the transistor is connected in common emitter mode, the current gain $\alpha =\frac{I_C}{I_B}$.
Figure shows a NPN transistor connected in the common base and common emitter configurations. Since, $95$% of the injected electrons reach the collector, the collector current is almost equal to the emitter current. Almost all transistors have a, in the range $0.95$ to $0.99$.
We know that
$\alpha =\frac{I_C}{I_E}=\frac{I_C}{I_B+I_C}(\because I_E=I_g+I_C)$

$\frac{1}{\alpha }=\frac{I_B+I_C}{I_C}=\frac{I_B}{I_C}+1$

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

$\therefore \beta =\frac{\alpha }{1-\alpha }$

Usually $\beta$ lies between $50$ and $300$. Some transistors have $\beta$ as high as $1000$.