The relation between `alpha` and `beta` parameters of current gains for a transistors is given by

Options

A. `beta = (1+alpha)/(beta)`
B. `alpha = (beta)/(1+beta)`
C. `alpha = (beta)/(1-beta)`
D. `beta = (alpha)/(1+alpha)`

Answer

To derive the relationship between the alpha (α) and beta (β) parameters of current gains for transistors, we can follow these steps:

### Step-by-Step Solution:

1. **Define the Parameters**:
- The current gain in the common base configuration is represented by alpha (α), which is defined as:
\[ \alpha = \frac{I_C}{I_E} \]
- The current gain in the common emitter configuration is represented by beta (β), defined as:
\[ \beta = \frac{I_C}{I_B} \]

2. **Understand the Current Relationships**:
- In a transistor, the emitter current (I_E) is the sum of the base current (I_B) and the collector current (I_C):
\[ I_E = I_B + I_C \]

3. **Substituting for I_E**:
- We can express α in terms of I_B and I_C:
\[ \alpha = \frac{I_C}{I_B + I_C} \]

4. **Rearranging the Equation**:
- To relate α to β, we can express I_B in terms of I_C using β:
\[ I_B = \frac{I_C}{\beta} \]
- Substitute this expression for I_B back into the equation for α:
\[ \alpha = \frac{I_C}{\frac{I_C}{\beta} + I_C} \]

5. **Simplifying the Expression**:
- Factor out I_C from the denominator:
\[ \alpha = \frac{I_C}{I_C \left(\frac{1}{\beta} + 1\right)} = \frac{1}{\frac{1}{\beta} + 1} \]

6. **Finding a Common Denominator**:
- Rewrite the denominator:
\[ \alpha = \frac{1}{\frac{1 + \beta}{\beta}} = \frac{\beta}{1 + \beta} \]

7. **Final Relationship**:
- Thus, the relationship between α and β is given by:
\[ \alpha = \frac{\beta}{1 + \beta} \]

### Summary:
The relationship between the alpha (α) and beta (β) parameters of current gains for transistors is:
\[ \alpha = \frac{\beta}{1 + \beta} \]