The rms value of a carrier voltage is 100 volts. Compute its rms value when it has been amplitude modulated by a sinusoidal audio voltage to a depth of 30%.

Options

A. 94V
B. 104.5V
C. 114.4V
D. 124V

Explanation

To solve the problem of finding the RMS value of a carrier voltage after amplitude modulation, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Given Values**:
- The RMS value of the carrier voltage (Vc) is given as 100 volts.
- The modulation index (m) is given as 30%, which can be expressed as a decimal:
\[ m = 0.3 \]

2. **Understand the Formula for RMS Value After Amplitude Modulation**:
- The formula for the total RMS voltage (Vrms) after amplitude modulation is:
\[ V_{rms} = V_c \sqrt{1 + \frac{m^2}{2}} \]
- Here, \(V_c\) is the RMS value of the carrier voltage, and \(m\) is the modulation index.

3. **Substitute the Known Values into the Formula**:
- Substitute \(V_c = 100\) volts and \(m = 0.3\) into the formula:
\[ V_{rms} = 100 \sqrt{1 + \frac{(0.3)^2}{2}} \]

4. **Calculate the Modulation Factor**:
- Calculate \(m^2\):
\[ m^2 = (0.3)^2 = 0.09 \]
- Now compute \(\frac{m^2}{2}\):
\[ \frac{m^2}{2} = \frac{0.09}{2} = 0.045 \]

5. **Complete the Calculation Inside the Square Root**:
- Now add 1 to the result:
\[ 1 + 0.045 = 1.045 \]

6. **Calculate the Square Root**:
- Find the square root of 1.045:
\[ \sqrt{1.045} \approx 1.022 \]

7. **Final Calculation for Vrms**:
- Multiply by the carrier voltage:
\[ V_{rms} = 100 \times 1.022 \approx 102.2 \text{ volts} \]

### Conclusion:
The RMS value of the modulated signal is approximately 102.2 volts.