Question

If the maximum amplitude of a modulated wave is three times its minimum amplitude, then the modulation index of the wave is

• A. 0.55
• B. 0.25
• C. 0.75
• D. 0.5

Hint:

The modulation index ($\mu$) is a crucial parameter in AM, indicating the extent of amplitude variation. Remember its formula in terms of maximum and minimum amplitudes: $\mu = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}}$. This formula is highly useful for problems involving peak amplitudes of modulated waves. The modulation index for AM is typically between 0 and 1.

Answer 1

The Correct Option is D

Step 1: Understand Amplitude Modulation and Define Amplitudes
In amplitude modulation (AM), the amplitude of the carrier wave varies in accordance with the amplitude of the modulating signal.
Let $A_{\text{max}}$ be the maximum amplitude of the modulated wave.
Let $A_{\text{min}}$ be the minimum amplitude of the modulated wave.
The problem states that the maximum amplitude of a modulated wave is three times its minimum amplitude. So, we can write: \[ A_{\text{max}} = 3 \times A_{\text{min}} \quad \cdots (1) \] Step 2: Recall the Formula for Modulation Index
The modulation index ($\mu$) for an amplitude-modulated wave is defined as the ratio of the change in amplitude of the carrier wave to the unmodulated carrier wave amplitude. It can also be expressed in terms of the maximum and minimum amplitudes of the modulated wave as: \[ \mu = \frac{A_{\text{max}} - A_{\text{min}}}{A_{\text{max}} + A_{\text{min}}} \] Step 3: Substitute the Given Relationship into the Modulation Index Formula
Substitute $A_{\text{max}} = 3 A_{\text{min}}$ from equation (1) into the modulation index formula: \[ \mu = \frac{(3 A_{\text{min}}) - A_{\text{min}}}{(3 A_{\text{min}}) + A_{\text{min}}} \] Simplify the numerator and the denominator: \[ \mu = \frac{2 A_{\text{min}}}{4 A_{\text{min}}} \] Cancel out $A_{\text{min}}$ from the numerator and denominator: \[ \mu = \frac{2}{4} \] \[ \mu = 0.5 \] Step 4: Analyze Options
\begin{itemize} \item Option (1): 0.55. Incorrect. \item Option (2): 0.25. Incorrect. \item Option (3): 0.75. Incorrect. \item Option (4): 0.5. Correct, as it matches our calculated modulation index. \end{itemize}