Question

An audio signal \( 10 \sin 2\pi(1500)t \) volt amplitude modulates a carrier \( 40 \sin 2\pi(10^5)t \) volts. The modulation factor and percentage modulation are:

• A. \( 0.25, 25\% \)
• B. \( 0.40, 40\% \)
• C. \( 0.10, 10\% \)
• D. \( 0.50, 50\% \)

Hint:

Always ensure you are using peak amplitudes for \( A_m \) and \( A_c \); if the signal is given in root-mean-square (RMS) values, you must convert them to peak values first, though the ratio will remain the same.

Answer 1

The Correct Option is A

Step 1: Recall the modulation factor.
In amplitude modulation, the modulation factor $\mu$ compares how strong the message is against the carrier, $\mu = \dfrac{A_m}{A_c}$, where $A_m$ is the audio amplitude and $A_c$ the carrier amplitude.
Step 2: Read off the audio amplitude.
The audio signal is $10 \sin 2\pi(1500)t$, so its peak amplitude is $A_m = 10$ V.
Step 3: Read off the carrier amplitude.
The carrier is $40 \sin 2\pi(10^5)t$, so its peak amplitude is $A_c = 40$ V.
Step 4: Form the ratio.
\[ \mu = \frac{A_m}{A_c} = \frac{10}{40} = 0.25 \]
Step 5: Convert to a percentage.
Percentage modulation is just $\mu \times 100\%$, so $0.25 \times 100\% = 25\%$.
Step 6: Match the option.
The factor is $0.25$ and the percentage is $25\%$, which is option 1.
\[ \boxed{0.25, \ 25\%} \]