The relationship between `alpha` and `beta` is given by

Options

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

Answer

To find the relationship between the parameters alpha (α) and beta (β) in a transistor, we can follow these steps:

### Step 1: Define the parameters
- **Alpha (α)** is defined as the ratio of the collector current (IC) to the emitter current (IE):
\[ \alpha = \frac{I_C}{I_E} \]
- **Beta (β)** is defined as the ratio of the collector current (IC) to the base current (IB):
\[ \beta = \frac{I_C}{I_B} \]

### Step 2: Relate the currents in a transistor
In a transistor, the emitter current (IE) is the sum of the collector current (IC) and the base current (IB):
\[ I_E = I_C + I_B \]

### Step 3: Express IB in terms of IC and IE
From the equation \(I_E = I_C + I_B\), we can rearrange it to find IB:
\[ I_B = I_E - I_C \]

### Step 4: Substitute IB into the beta equation
Now, we can substitute the expression for IB into the beta equation:
\[ \beta = \frac{I_C}{I_B} = \frac{I_C}{I_E - I_C} \]

### Step 5: Express β in terms of α
Next, we can express \(I_C\) in terms of \(I_E\) and α:
\[ I_C = \alpha I_E \]
Substituting this into the equation for β gives:
\[ \beta = \frac{\alpha I_E}{I_E - \alpha I_E} \]

### Step 6: Simplify the equation
Now, simplify the denominator:
\[ \beta = \frac{\alpha I_E}{I_E(1 - \alpha)} = \frac{\alpha}{1 - \alpha} \]

### Step 7: Rearranging to find the relationship
From the equation above, we can rearrange it to find the relationship between α and β:
\[ \beta (1 - \alpha) = \alpha \]
This can be rewritten as:
\[ \beta - \beta \alpha = \alpha \]
Thus, we can express the relationship as:
\[ \beta = \frac{\alpha}{1 - \alpha} \]

### Final Relationship
The final relationship between α and β is:
\[ \beta = \frac{\alpha}{1 - \alpha} \]