If a number of sine waves with modulation indices `n_(1),n_(2),n_(3)`… modulate a carrier wave, then total modulation index (n) of the wave is

Options

A. `n_(1)+n_(2)…. +2(n_(1)+n_(2)…..)`
B. `sqrt(n_(1)-n_(2)+n_(3)…..)`
C. `sqrt(n_(1)^(2)+n_(2)^(2)+n_(3)^(2)….)`
D. none of these

Answer

To solve the problem of finding the total modulation index (n) when multiple sine waves with modulation indices \( n_1, n_2, n_3, \ldots \) modulate a carrier wave, we can follow these steps:

### Step-by-Step Solution:

1. **Understand Modulation Index**:
The modulation index (n) is defined as the ratio of the amplitude of the modulating signal (message wave) to the amplitude of the carrier wave. Mathematically, it can be expressed as:
\[ n = \frac{A_m}{A_c} \]
where \( A_m \) is the amplitude of the message wave and \( A_c \) is the amplitude of the carrier wave.

2. **Identify Individual Modulation Indices**:
For multiple sine waves, each sine wave has its own modulation index:
- \( n_1 = \frac{A_{m1}}{A_c} \)
- \( n_2 = \frac{A_{m2}}{A_c} \)
- \( n_3 = \frac{A_{m3}}{A_c} \)
- And so on for any number of sine waves.

3. **Combine Modulation Indices**:
When multiple sine waves modulate a carrier wave, the total modulation index is derived from the individual modulation indices. The relationship follows a square law, which means:
\[ n_{total} = \sqrt{n_1^2 + n_2^2 + n_3^2 + \ldots} \]

4. **Final Expression**:
Thus, the total modulation index can be expressed as:
\[ n = \sqrt{n_1^2 + n_2^2 + n_3^2 + \ldots} \]

5. **Conclusion**:
This formula gives us the resultant modulation index when multiple sine waves modulate a single carrier wave.