WILLER ACADEMY

Communication Systems

Class 12 Physics - Essential Formulas with Definitions & Applications

Chapter: 15 (CBSE Physics)

Board Exam Focus

Numerical Formulas

Modulation Index (μ)

Formula 1

\( \mu = \frac{A_m}{A_c} \)

Definition

Measures the extent of amplitude variation in AM waves. It is the ratio of modulating signal amplitude (\(A_m\)) to carrier signal amplitude (\(A_c\)).

Application & Constraints

Must be ≤ 1 to avoid distortion. Critical for determining signal quality and transmission efficiency.

Example

If \(A_m = 0.8V\) and \(A_c = 2V\), then \( \mu = \frac{0.8}{2} = 0.4 \) (40% modulation)

Total AM Power

Formula 2

\( P_t = P_c \left(1 + \frac{\mu^2}{2}\right) \)

Definition

Total transmitted power in amplitude modulation, including carrier and sidebands. \(P_c\) is carrier power (\(P_c = \frac{A_c^2}{2R}\), R = resistance).

Application

Used to calculate power distribution in AM transmission systems and transmitter design.

Example

For \( \mu = 0.6 \) and \( P_c = 100W \), \( P_t = 100 \left(1 + \frac{0.36}{2}\right) = 118W \)

Bandwidth of AM Signal

Formula 3

\( \text{BW} = 2f_m \)

Definition

Range of frequencies occupied by the AM wave. Equal to twice the highest modulating frequency (\(f_m\)).

Application

Essential for spectrum allocation in radio broadcasting and communication systems.

Example

For voice signal (up to \(f_m = 4\text{kHz}\)), BW = 8 kHz (standard AM radio bandwidth)

Signal-to-Noise Ratio

Formula 4

\( \text{SNR} = \frac{S}{N} \)

Definition

Ratio quantifying signal clarity, where \(S\) is signal power and \(N\) is noise power.

Application

Critical in receiver design to minimize noise interference. Higher SNR = better quality.

Note

SNR is often expressed in decibels: \( \text{SNR}_{\text{dB}} = 10 \log_{10}\left(\frac{S}{N}\right) \)

Shannon's Channel Capacity

Formula 5

\( C = B \log_2(1 + \text{SNR}) \)

Definition

Theoretical maximum data rate a channel can support without error, where \(B\) is bandwidth.

Application

Determines fundamental limits of communication systems (discovered by Claude Shannon in 1948).

Example

For \( B = 3\text{kHz} \), SNR = 100, \( C \approx 20\text{kbps} \)

AM Wave Equation

Formula 6

\( s(t) = A_c[1 + \mu \cos(2\pi f_m t)] \cos(2\pi f_c t) \)

Definition

Time-domain equation of AM wave combining carrier (\(f_c\)) and modulating (\(f_m\)) signals.

Application

Fundamental for understanding AM waveform characteristics and demodulation techniques.

📊 Formula Summary

Modulation Index: \( \mu = \frac{A_m}{A_c} \)

Total AM Power: \( P_t = P_c (1 + \frac{\mu^2}{2}) \)

Bandwidth: \( BW = 2f_m \)

Channel Capacity: \( C = B \log_2(1 + SNR) \)

⚠️ Exam Pitfalls

• Modulation index μ > 1 causes distortion

• Confusing sideband frequencies

• Inconsistent units (kHz vs MHz)

• Forgetting noise impact on SNR

📝 Board Exam Tips

• Always mention units in calculations

• Derive AM wave equation (5 marks)

• Sketch frequency spectrum diagrams

• Practice numericals on modulation index

WILLER ACADEMY

Class 12 Physics - Communication Systems | Chapter 15

Board Exam Preparation Resource | Numerical Formulas with Definitions