QB_PH_SEC8_013: Semiconductor Devices & Diodes

For a silicon BJT operating in the active region, the common-base current gain is $\alpha = 0.980$. What is the common-emitter current gain $\beta$?

Correct Answer

### Step-by-Step Derivation & Verification: 1. **Current Conservation in BJT**: $I_E = I_B + I_C$ 2. **Gain Definitions**: - Common-base current gain: $\alpha = \frac{I_C}{I_E}$ - Common-emitter current gain: $\beta = \frac{I_C}{I_B}$ 3. **Algebraic Relationship**: $\frac{1}{\alpha} = \frac{I_E}{I_C} = \frac{I_B + I_C}{I_C} = \frac{1}{\beta} + 1$ $\frac{1}{\beta} = \frac{1}{\alpha} - 1 = \frac{1 - \alpha}{\alpha} \implies \beta = \frac{\alpha}{1 - \alpha}$ 4. **Numerical Substitution**: $\beta = \frac{0.980}{1 - 0.980} = \frac{0.980}{0.020} = 49$ 5. **Official Exact Integer**: 49.