Chapter 15: Communication Systems - Comprehensive NEET Physics Formulae Summary

1. Key Formulae and Explanations

1. Speed of Electromagnetic Waves: $c=\nu \lambda$
2. • Explanation: This formula represents the speed of electromagnetic waves, where $c$ is the speed of light (approximately $3\times 10^8,\text{m/s}$ in a vacuum), $\nu$ is the frequency of the wave, and $\lambda$ is the wavelength. It is fundamental in understanding wave propagation, particularly in communication systems.
2. Range of Line-of-Sight Communication: $d_M=\sqrt{2R{h}_T}+\sqrt{2R{h}_R}$
3. • Explanation: This formula calculates the maximum line-of-sight distance $d_M$ between two antennas, where $h_T$ and $h_R$ are the heights of the transmitting and receiving antennas, respectively, and $R$ is the Earth's radius. It is crucial for determining the coverage area in communication systems like TV and radio broadcasting.
3. Modulation Index for Amplitude Modulation (AM): $\mu =\frac{A_m}{A_c}$
4. • Explanation: The modulation index $\mu$ is the ratio of the amplitude of the message signal $A_m$ to the amplitude of the carrier signal $A_c$. It quantifies the extent of modulation applied to the carrier wave. For efficient transmission, $\mu$ should be kept between 0 and 1.
4. Amplitude Modulated Wave Equation: $c_m(t)=\left(A_c+A_m\sin {\omega }_{m}t\right)\sin {\omega }_{c}t$
5. • Explanation: This equation represents an amplitude-modulated wave, where $A_c$ and ${\omega }_c$ are the amplitude and angular frequency of the carrier wave, and $A_m$ and ${\omega }_m$ are the amplitude and angular frequency of the message signal. It shows how the carrier wave's amplitude varies in response to the message signal.
5. Frequency of Sidebands in AM: $\text{Upper Sideband}={\omega }_c+{\omega }_m$, $\text{Lower Sideband}={\omega }_c-{\omega }_m$
6. • Explanation: In amplitude modulation, the transmitted signal generates two sidebands, upper and lower, around the carrier frequency. These sidebands carry the information from the message signal. The frequencies of these sidebands are the sum and difference of the carrier frequency ${\omega }_c$ and the message signal frequency ${\omega }_m$.
6. Bandwidth of Amplitude Modulated Signal: $B=2f_m$
7. • Explanation: The bandwidth $B$ required for transmitting an amplitude-modulated signal is twice the highest frequency of the modulating signal $f_m$. This is critical in allocating frequency spectrum for AM transmission.

2. Derivations of Important Formulae

• Derivation of the Amplitude Modulated Wave: The amplitude modulated wave is derived by combining the carrier wave and the modulating signal. Starting with: $c_m(t)=\left(A_c+A_m\sin {\omega }_{m}t\right)\sin {\omega }_{c}t$ Using trigonometric identities, this expands to: $c_m(t)=A_c\sin {\omega }_{c}t+\frac{A_m}{2}\left[\cos ({\omega }_c-{\omega }_m)t-\cos ({\omega }_c+{\omega }_m)t\right]$ This shows the presence of the carrier frequency and the sidebands.

3. Example Applications

Example Problem: Given a carrier wave of frequency 1 MHz and a message signal of frequency 10 kHz with a peak voltage of 5V, determine the modulation index and the frequencies of the sidebands.

Solution:

1. Modulation Index: $\mu =\frac{5}{10}=0.5$
2. Sideband Frequencies:
3. • Upper Sideband: $1\text{MHz}+10\text{kHz}=1.01\text{MHz}$
   • Lower Sideband: $1\text{MHz}-10\text{kHz}=0.99\text{MHz}$

4. Common Mistakes and Tips

• Common Mistake: Confusing the modulation index with the percentage of modulation.
• • Tip: Remember that the modulation index $\mu$ is a ratio and should be between 0 and 1. A value of $\mu =0.75$ corresponds to 75% modulation.
• Common Mistake: Forgetting to include both sidebands in the bandwidth calculation.
• • Tip: Always double the maximum message signal frequency to determine the bandwidth required for AM.

5. Final Review and Formatting

Ensure all formulae are clearly written with appropriate explanations. The examples provided should reflect common NEET problems, with step-by-step solutions that illustrate the correct application of each formula. Consistent formatting with proper use of symbols and units is crucial for clarity and quick revision.

This summary of formulae from the chapter on Communication Systems covers the essential equations and their applications, providing a strong foundation for NEET UG Physics preparation.