Question

For \(100 \%\) modulation, What is the value of minimum amplitude \(\left(E_{\min }\right)\) of the modulated wave? (A) Zero (B) 1 (C) 2 (D) 3

Short answer

(B) 1

Step-by-step solution

Write down the given information

We are given that the modulation index, m, is 100% or 1. Our goal is to find the minimum amplitude, \(E_{min}\), of the modulated wave. #Step 2: Solve for E_min using the modulation index formula#

Solve for E_min using the modulation index formula

Since we know that \(m = 1\), we can rearrange the modulation index formula to solve for \(E_{min}\): \( 1 = \frac{A_c - E_{min}}{A_c + E_{min}} \) Now we need to isolate \(E_{min}\) on one side of the equation. To do so, first multiply both sides by \(A_c + E_{min}\): \( A_c + E_{min} = A_c - E_{min} \) Next, add \(E_{min}\) to both sides: \( 2E_{min} = A_c \) Now, divide both sides by 2 to get: \( E_{min} = \frac{A_c}{2} \) Since the minimum amplitude, \(E_{min}\), is half of the amplitude of the carrier wave, \(A_c\), this means that it must be some nonzero value. Comparing this result with the given options, we find that the correct answer is: (B) 1

Key concepts

Modulation Index

In amplitude modulation, the modulation index is a crucial parameter. It describes the extent to which the amplitude of the carrier wave is varied by the message signal.
It essentially determines the depth of modulation in the carrier signal. To put it simply, the modulation index, denoted as \( m \), is a ratio that compares the message signal amplitude to the carrier signal amplitude.
If the modulation index is 1 or 100%, it means the message signal perfectly complements the carrier wave, achieving full modulation.

• When \( m = 1 \), the amplitude variation reaches its maximum, leading to 100% modulation.
• If \( m < 1 \), the modulation is under-modulated, meaning the message signal does not fully modulate the carrier.
• Conversely, if \( m > 1 \), over-modulation occurs, which can distort the signal.

The modulation index formula is used to find the minimum and maximum amplitude variations in the modulated wave. This provides a clear picture of how the message signal will affect the carrier wave.

Minimum Amplitude

Understanding minimum amplitude in the context of a modulated wave is essential. The minimum amplitude of a modulated wave, denoted as \( E_{min} \), signifies the lowest point of amplitude in the modulating signal.
It is directly influenced by the modulation index and the carrier wave amplitude.
From the modulation index formula, we find that for 100% modulation \((m = 1)\), the minimum amplitude is given by the equation:
\[ E_{min} = \frac{A_c}{2} \]

• This indicates that at full modulation, the minimum amplitude is exactly half of the carrier amplitude \( A_c \).
• At this level, the waveform reaches its most compressed state.

Knowing \( E_{min} \) helps ensure that the modulated signal does not drop to zero, which would result in loss of information.

Carrier Wave Amplitude

Carrier wave amplitude, represented by \( A_c \), is an integral part of amplitude modulation. This amplitude indicates the strength or level of the carrier wave that is modulated by the message signal.
The size of \( A_c \) determines the baseline amplitude around which modulation occurs.
Consider \( A_c \) the centerpiece of modulation:

• It serves as the constant amplitude before any modulation.
• Its value remains constant under ideal conditions if there is no amplitude variance by modulation effects.
• During modulation, the amplitude of the carrier wave is modified according to the characteristics of the message signal, leading to a new range of amplitudes between \( E_{min} \) and maximum amplitude \( E_{max} \).

Understanding the carrier wave amplitude is crucial as it guides the modulation process by providing the reference level from which amplitudes are varied.