JEE Physics Communication System Notes Set 02

Modulation and Bandwidth

A message signal of low frequency is superimposed on a high-frequency carrier so that antennas can be small (about a quarter of a wavelength), power radiated is effective and many stations can share the spectrum. In amplitude modulation the carrier amplitude follows the message: \( c_m(t)=(A_c+A_m\sin\omega_mt)\sin\omega_ct \), with modulation index \( \mu=\frac{A_m}{A_c}=\frac{A_{max}-A_{min}}{A_{max}+A_{min}} \), which should not exceed 1. The modulated wave contains the carrier and two side bands at \( f_c\pm f_m \), so the bandwidth is \( 2f_m \). In frequency modulation the carrier frequency follows the message, and it is more immune to noise. Modulation does not shorten the time lag of the signal.

Propagation of Radio Waves

Ground waves serve frequencies up to a few MHz, sky waves (reflected by the ionosphere) serve roughly 3 to 30 MHz, and space waves (line of sight, satellites) serve frequencies above about 30 MHz, since these pass through the ionosphere. The line-of-sight range between two antennas of heights \( h_T \) and \( h_R \) is \( d=\sqrt{2Rh_T}+\sqrt{2Rh_R} \), and a single antenna of height h covers \( \sqrt{2Rh} \). Sky wave signals are stronger at night. For any electromagnetic wave, \( E_0=cB_0 \) and \( \lambda=c/f \), and the number of photons emitted per second equals the power divided by \( hf \).

Worked Examples

A 12 V carrier modulated to 75 percent has a message amplitude of 9 V.
A carrier of 1 MHz modulated by 10 kHz gives side bands at 1010 kHz and 990 kHz.
Antennas at heights 32 m and 50 m give a range of about 20.2 + 25.3 = 45.5 km.