BJT Beta (β) Calculation from Alpha (α)

This solution explains how to find the value of the common-emitter current gain, β, for a Bipolar Junction Transistor (BJT) given its common-base current gain, α.

Understanding BJT Current Gain Parameters

Bipolar Junction Transistors (BJTs) have key parameters that describe their current amplifying capabilities:

• α (Alpha): This represents the common-base current gain. It is defined as the ratio of the change in collector current ($\Delta I_C$) to the change in emitter current ($\Delta I_E$) while keeping the collector-base voltage ($V_{CB}$) constant. It's always less than 1.

  Formula: $\alpha = \frac{\Delta I_C}{\Delta I_E}$

• β (Beta): This represents the common-emitter current gain. It is defined as the ratio of the change in collector current ($\Delta I_C$) to the change in base current ($\Delta I_B$) while keeping the collector-emitter voltage ($V_{CE}$) constant. It's typically much larger than 1.

  Formula: $\beta = \frac{\Delta I_C}{\Delta I_B}$

The Relationship Between α and β

For any active BJT, there's a direct relationship between α and β. The formula connecting them is:

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

Conversely, if you know β, you can find α using:

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

Calculating Beta (β) for the Given Alpha (α)

We are given the value of α = 0.99.

To find β, we use the formula:

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

Let's substitute the given value of α:

1. Substitute $\alpha = 0.99$ into the formula:

   $$ \beta = \frac{0.99}{1 - 0.99} $$

2. Calculate the denominator:

   $$ 1 - 0.99 = 0.01 $$

3. Perform the division:

   $$ \beta = \frac{0.99}{0.01} $$

   $$ \beta = 99 $$

Final Result

The calculated value of β for the BJT is 99. This value corresponds to the first option provided.