Question

A transmitter transmit a power of $10\mathrm{KW}$ when modulation is $50\mathrm{\%}$. What is the power of carrier wave? (A) $12\mathrm{KW}$ (B) $7.0\mathrm{KW}$ (C) $8.89\mathrm{KW}$ (D) $5.0\mathrm{KW}$

Short answer

The power of the carrier wave is (B) $7.0\mathrm{KW}$.

Step-by-step solution

Understanding modulation in Amplitude Modulation

Amplitude Modulation (AM) is a modulation technique where the amplitude of the carrier wave is varied in proportion to the amplitude of the modulating (information) signal. The radio frequency carrier wave has a much higher frequency than that of the modulating (information) signal, and the modulation index is a measure of how much the amplitude of the carrier wave is varied by the modulating signal.

Apply the modulation index formula

In Amplitude Modulation, the total power Pt of the modulated signal is given by the formula: $Pt=Pc(1+{\displaystyle \frac{m^2}{2}})$, where Pt is the total transmitted power, Pc is the power of the carrier wave, m is the modulation index. The modulation index 'm' is expressed in percentage, and in this case, it is given as 50% (0.5 in decimal). Having this formula and the necessary values, we can then solve for the power of the carrier wave, Pc.

Determine the carrier wave power

Given the total transmitted power Pt = 10 KW and modulation index m = 0.5, we can now solve for the carrier wave power Pc: $10=Pc(1+{\displaystyle \frac{(0.5{)}^2}{2}})$ After solving for Pc, we get: Pc = $7.0KW$ Answer: The power of the carrier wave is (B) $7.0\mathrm{KW}$.

Key concepts

Understanding the Modulation Index

The modulation index is a fundamental concept in Amplitude Modulation (AM). It refers to the extent of variation in the carrier wave caused by the modulating signal. Expressed as a ratio or percentage, it quantifies the modulation depth. Mathematically, the modulation index, denoted by 'm,' is defined as the ratio of the peak change in amplitude of the modulating signal to the amplitude of the carrier wave.
For a 50% modulation index, the value of 'm' would be 0.5 in decimal. A higher modulation index indicates more substantial variation in carrier amplitude, while a lower index points to smaller changes. Crucially:

• A modulation index greater than 1 leads to overmodulation, which can cause distortion.
• A modulation index of 1 signifies 100% modulation and is typically the maximum permissible level without distortion in standard AM.

Understanding the modulation index helps us adjust the broadcasting strength and quality of an Amplitude Modulated signal effectively.

What is a Carrier Wave?

In Amplitude Modulation, the carrier wave serves as the main vehicle for transmitting information. This wave is a high-frequency sine wave that does not carry any useful information by itself. Instead, its amplitude is modified in accordance with the modulating signal, which contains the information to be transmitted.
The characteristics of a carrier wave include:

• High Frequency: Allows the wave to travel long distances efficiently.
• Constant Amplitude: In its unmodulated form, the carrier wave has a steady magnitude.
• Bright Stability: Provides a stable 'base' for modulation.

Through Amplitude Modulation, the carrier wave is transformed, enabling the human voice or music to be carried over vast distances in a format that can be easily transmitted and received by radio systems.

Calculating Transmitted Power in AM

Transmitted power is an essential aspect of Amplitude Modulation that combines both the carrier and modulated signals. In an AM system, the total transmitted power is derived from both the power of the carrier wave (Pc) and the power introduced by the modulation.
The relationship between these comes through the formula:$$Pt=Pc(1+\frac{m^2}{2})$$Where:

• $Pt$: Total transmitted power.
• $Pc$: Carrier wave power.
• $m$: Modulation index.

This formula shows how the modulation index (m) affects the total transmitted power. With the given modulation index and total transmitted power in a system, one can solve for the unknown parameter, typically the carrier wave power. Solving such equations allows engineers to optimize and achieve the right balance in a system, ensuring the signal is clear and efficient without being overpowered or distorted.