if the maximum and minimum voltage of AM wave are `V_(max)` and `V_(min)`, respectively then modulation factor

Answer

To find the modulation factor for an AM wave given the maximum voltage \( V_{\text{max}} \) and minimum voltage \( V_{\text{min}} \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The **carrier wave** voltage \( V_c \) is the average voltage of the AM wave.
- The **message wave** voltage \( V_m \) is the difference between the maximum and minimum voltages of the AM wave.

2. **Calculate the Average (Carrier Wave) Voltage**:
\[ V_c = \frac{V_{\text{max}} + V_{\text{min}}}{2} \]

3. **Calculate the Amplitude (Message Wave) Voltage**:
\[ V_m = \frac{V_{\text{max}} - V_{\text{min}}}{2} \]

4. **Define the Modulation Factor**:
The modulation factor \( m \) is defined as the ratio of the amplitude of the message wave to the average voltage of the carrier wave:
\[ m = \frac{V_m}{V_c} \]

5. **Substitute the Values**:
Substitute the expressions for \( V_m \) and \( V_c \) into the modulation factor equation:
\[ m = \frac{\frac{V_{\text{max}} - V_{\text{min}}}{2}}{\frac{V_{\text{max}} + V_{\text{min}}}{2}} \]

6. **Simplify the Expression**:
The \( \frac{1}{2} \) terms cancel out:
\[ m = \frac{V_{\text{max}} - V_{\text{min}}}{V_{\text{max}} + V_{\text{min}}} \]

7. **Final Result**:
Thus, the modulation factor \( m \) is given by:
\[ m = \frac{V_{\text{max}} - V_{\text{min}}}{V_{\text{max}} + V_{\text{min}}} \]