Question

A signal \(\mathrm{e}_{\mathrm{m}}=20 \sin 2000 \pi \mathrm{t}\) amplitude modulates a carries wave \(e_{c}=80 \sin 200000 \pi t\). The percentage of modulation is (A) \(25 \%\) (B) \(0.25 \%\) (C) \(400 \%\) (D) \(40 \%\)

Short answer

The percentage of modulation is calculated using the modulation index, which is the ratio of the amplitude of the modulating signal \(E_m\) to the amplitude of the carrier signal \(E_c\). In this case, \(E_m = 20\) and \(E_c = 80\), so the modulation index is \(\frac{E_m}{E_c} = \frac{20}{80} = \frac{1}{4}\). To convert the modulation index to a percentage, multiply it by 100: \(Percentage\,of\,modulation = \frac{1}{4} \times 100\% = 25\%\). Therefore, the correct answer is (A) 25%.

Step-by-step solution

Define the modulating and carrier signal

The modulating signal is given by the equation: \(e_m = 20 \sin(2000 \pi t)\) and the carrier signal is given by the equation: \(e_c = 80 \sin(200000 \pi t)\).

Calculate the amplitude ratio

The modulation index \(m\) is defined as the ratio of the amplitude of the modulating signal \(E_m\) to the amplitude of the carrier signal \(E_c\): \[m = \frac{E_m}{E_c}\] Now we need to find the amplitude of both signals. From the equations, the amplitudes are as follows: \(E_m = 20\) and \(E_c = 80\) Now we can find the modulation index: \[m = \frac{20}{80}\]

Calculate the modulation index

Using the amplitude values from step 2, we have: \[m = \frac{20}{80} = \frac{1}{4}\]

Convert the modulation index to a percentage

To convert the modulation index to a percentage, multiply it by 100: \[Percentage\,of\,modulation = m \times 100\%\] Now, substituting the modulation index value from step 3: \[Percentage\,of\,modulation = \frac{1}{4} \times 100\% = 25\%\] The correct answer is (A) 25%.

Key concepts

Modulating Signal

In the world of amplitude modulation, the modulating signal plays a crucial role as it carries the information we wish to communicate. This signal is oftentimes an audio signal, such as voice or music, and it has specific characteristics that define its behavior.
In our example exercise, the modulating signal is represented by the equation \(e_m = 20 \sin(2000 \pi t)\).
Here are a few important aspects to understand about this signal:

• Amplitude: The amplitude of the modulating signal is 20. This indicates the maximum value that the signal can reach and directly affects the modulation process.
• Frequency: The frequency is determined by the term \(2000 \pi t\), which upon simplification, becomes 1000 Hz. This tells us how many cycles the signal completes in one second.

The modulating signal changes over time and gets superimposed onto another wave: the carrier wave. This is how information is transmitted through modulation.

Carrier Wave

The carrier wave serves as the backbone on which the modulating signal is imposed, and it controls the fundamental properties of the transmitted signal. It is typically a high-frequency signal, ensuring that the modulated waveform can travel long distances.
In our given exercise, the carrier wave is expressed as \(e_c = 80 \sin(200000 \pi t)\). Let's examine this wave in more detail:

• Amplitude: The amplitude here is 80, much higher than the modulating signal's amplitude. This greater amplitude makes it possible for the signal to be efficiently transmitted over longer distances.
• Frequency: When broken down, the frequency emerges as 100,000 Hz or 100 kHz. This high frequency corresponds to the rapid oscillations of the carrier wave. It's much higher than the modulating signal's frequency, fulfilling the requirement for effective modulation.

The carrier wave acts as a neutral medium that allows the modulating signal to hitch a ride, carrying it to its destination intact and without distortion.

Modulation Index

The modulation index is vital in defining how much the modulating signal affects the carrier wave. It is the measure of the extent of modulation used in the transmission process.
The modulation index \(m\) is calculated as the ratio of the amplitude of the modulating signal \(E_m\) to that of the carrier wave \(E_c\). In our example, this calculation proves enlightening:

• \(E_m = 20\)
• \(E_c = 80\)
• \(m = \frac{20}{80} = \frac{1}{4}\)

The resulting modulation index of \(\frac{1}{4}\) or 0.25 indicates a relatively low level of modulation. To express this as a percentage, we multiply by 100, arriving at 25%.
This percentage demonstrates how much of the carrier wave's amplitude is varied due to the modulating signal. In this scenario, the variation is just enough to transmit the full range of information without causing distortion or interface during transmission.