Question

For an amplitude modulated wave, the maximum amplitude is found to be \(9 \mathrm{~V}\) while the minimum amplitude is found to be \(3 \mathrm{~V}\). The modulation index is (A) \(100 \%\) (B) \(75 \%\) (C) \(50 \%\) (D) \(25 \%\)

Short answer

The modulation index, given by the formula \(m = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}\), can be calculated by plugging in the given values: \(m = \frac{9 - 3}{9 + 3} = \frac{6}{12} = \frac{1}{2} = 0.5\). Converted to a percentage, the modulation index is \(0.5 * 100 = 50 \% \). Therefore, the correct answer is (C) 50%.

Step-by-step solution

Write down the formula for the modulation index

The formula to find the modulation index is: \(m = \frac{A_{max} - A_{min}}{A_{max} + A_{min}}\)

Plug in the values

We are given the values for \(A_{max}\) and \(A_{min}\). Now, we can plug these values into the formula: \(m = \frac{9 - 3}{9 + 3} \)

Calculate the modulation index

Solve the equation: \(m = \frac{6}{12} \)

Simplify the modulation index

Simplify the modulation index: \(m = \frac{1}{2}\), which can also be written as \(m = 0.5\)

Convert the modulation index to percentage

To convert the modulation index to percentage, multiply it by 100: \(m \% = 0.5 * 100 = 50 \% \) The modulation index is 50%, which corresponds to answer choice (C).

Key concepts

Amplitude Modulation

Amplitude Modulation, often abbreviated as AM, is a technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In this process, the amplitude of a carrier signal is varied in proportion to a modulating signal, such as an audio signal. This modulation allows the transmission of the desired information over long distances. AM is widely used because of its simplicity and the ease of implementation in analog devices. However, one of the main limitations of AM is that it is susceptible to noise and interference, which can degrade the quality of the received signal. Understanding how AM works is crucial for anyone studying communication systems, as it forms the foundation for more advanced modulation schemes.

Wave Amplitude

Wave amplitude refers to the maximum extent of a vibration or oscillation, measured from the position of equilibrium. In the context of amplitude modulation, the amplitude of the carrier wave is what is being varied to encode the information. The highest value this amplitude reaches is the maximum amplitude, while the lowest value is the minimum amplitude. For amplitude modulated waves, understanding the difference between maximum and minimum amplitudes is critical. This difference is directly related to the modulation index calculation, an essential parameter for determining the effectiveness and quality of the modulation. A good understanding of wave amplitude not only helps in comprehending AM signals but also aids in processing other types of wave-based communication signals.

Percentage Calculation

Percentage calculation is a basic mathematical concept often used to express a number as a fraction of 100. In the context of amplitude modulation, converting the modulation index to a percentage is crucial for interpreting the degree of modulation. The modulation index, found to be 0.5 in the original exercise, indicates that the carrier wave's amplitude reaches two-thirds of its maximum potential change. By converting this figure into a percentage, we get 50%, which tells us about the amplitude modulation level clearly. The conversion process involves a simple multiplication of the modulation index by 100 to get a percentage, which is more intuitive for expressing the extent of modulation efforts in communication systems.

Communication Systems

Communication systems are complex networks that involve the transmission of information from one place to another, employing various technologies and methodologies, including amplitude modulation. These systems are fundamental to modern connectivity and range from simple wireless radio transmissions to elaborate satellite networks. The choice of modulation technique, such as amplitude modulation for certain types of broadcasts, plays a critical role in determining the efficiency and quality of the communication system. Factors like the modulation index help engineers and designers understand how well the information is being transmitted and how much interference and noise can be tolerated without degrading the signal. As communication technologies evolve, understanding foundational methods like amplitude modulation paves the way for grasping more sophisticated systems, maintaining the essential groundwork upon which all modern communication rests.