BJT Current Gain Relationship: Alpha ($\alpha$) and Beta ($\beta$)

Understanding BJT Current Gains

A bipolar junction transistor (BJT) has two primary current gain parameters:

• Common-Base Current Gain ($\alpha$): Defined as the ratio of collector current ($I_C$) to emitter current ($I_E$). It is typically close to 1 (e.g., 0.95 to 0.99). $\alpha = \frac{I_C}{I_E}$
• Common-Emitter Current Gain ($\beta$): Defined as the ratio of collector current ($I_C$) to base current ($I_B$). It is typically much larger than 1 (e.g., 20 to 200). $\beta = \frac{I_C}{I_B}$

Deriving the Relationship

The relationship between these gains can be derived using Kirchhoff's Current Law at the emitter node: $I_E = I_C + I_B$ Using the definition of $\alpha$, we can write $I_C$ in terms of $I_E$: $I_C = \alpha \cdot I_E$ Substitute this into the KCL equation: $I_E = (\alpha \cdot I_E) + I_B$ Rearrange to solve for $I_B$: $I_B = I_E - (\alpha \cdot I_E)$ $I_B = I_E (1 - \alpha)$ Now, use the definition of $\beta$: $\beta = \frac{I_C}{I_B}$ Substitute $I_C = \alpha \cdot I_E$ and $I_B = I_E (1 - \alpha)$: $\beta = \frac{\alpha \cdot I_E}{I_E (1 - \alpha)}$ Cancel out $I_E$: $\beta = \frac{\alpha}{1 - \alpha}$

Final Relationship

The relationship between the common-emitter current gain ($\beta$) and the common-base current gain ($\alpha$) is: $\beta = \frac{\alpha}{1 - \alpha}$