Question

An amplitude modulated wave is represented by \(C_m(t) = 10(1 + 0.2 \cos 12560 t) \sin(111 \times 10^4 t)\) volts. The modulating frequency in kHz will be _________.

Hint:

In the AM equation \(A_c(1 + m \cos \omega_m t) \sin \omega_c t\), remember that \(\omega_m \ll \omega_c\). The lower frequency always belongs to the message/modulating signal.

Answer 1

Correct Answer: 2

Step 1: Understanding the Concept:
An amplitude modulated (AM) signal is expressed as \(C_m(t) = A_c (1 + m \cos \omega_m t) \sin \omega_c t\). The term inside the cosine represents the modulating signal (message signal).
Step 2: Key Formula or Approach:
The modulating frequency \(f_m\) is calculated from the angular modulating frequency \(\omega_m\) using the relation:
\[ f_m = \frac{\omega_m}{2\pi} \]
Step 3: Detailed Explanation:
Comparing the given equation with the standard AM equation:
\[ \omega_m = 12560 \text{ rad/s} \]
Calculating linear frequency \(f_m\):
\[ f_m = \frac{12560}{2 \times 3.14159} \]
Using \(\pi \approx 3.14\):
\[ f_m \approx \frac{12560}{6.28} = 2000 \text{ Hz} \]
Converting to kHz:
\[ f_m = \frac{2000}{1000} = 2 \text{ kHz} \]
Step 4: Final Answer:
The modulating frequency is 2 kHz.