Question

When the signal amplitude is $1.5$ times the carrier amplitude then the percent of modulation is:
A. $50\% $
B. $100\% $
C. $150\% $
D. $75\% $

Answer 1

Hint: The formula for the modulation index, relating the amplitude of the signal and the amplitude of the carrier frequency is applied. Since signal amplitude is $1.5$ times the carrier amplitude the signal amplitude is written in terms of amplitude carrier wave and substituted in the formula to get the percent of modulation.

Complete step by step answer:
 We are required to find the percent of modulation that is the rate of modulation that has taken place. This is given by a factor known as modulation index for an amplitude modulated wave. The modulation index is defined as the ratio of the change in amplitude of the carrier wave to the amplitude of the signal that is the original carrier wave. This factor indicates the change in amplitude change of the signal with respect to the carrier wave. The modulation index is given by the formula:
$\mu = \dfrac{{{A_m}}}{{{A_c}}}$ ---($1$)
Next, we need to extract the data given in the question. As per the given data:
${A_m} = 1.5 \times {A_c}$ ----($2$)

The above equation gives the relation between the amplitude of the signal wave that is the original wave and the carrier wave. It gives the amplitude in terms of the carrier wave.
Hence, by substituting equation ($2$) in ($1$) we get:
$\mu = \dfrac{{1.5 \times {A_c}}}{{{A_c}}}$
The two terms of ${A_c}$ in the above equation are cancelled to get:
$\mu = 1.5$
We need to find the amount of modulation that is the percent of modulation that has taken place. So the formula in equation ($1$) becomes:
$\mu = \dfrac{{{A_m}}}{{{A_c}}} \times 100\% $
Since we have already determined the value of $\mu $ to be $1.5$:
$ \Rightarrow \mu = 1.5 \times 100\% $
$ \therefore \mu = 150\% $
Therefore, the percent of modulation is $150\% $

Hence, option B is the correct option.

Additional information: Modulation is the process by which some characteristics like amplitude, frequency and phase angle of a high frequency carrier wave is varied with respect to the instantaneous value of the low frequency audio signal which is known as the modulating signal.

This process is done by superimposing a low frequency information signal onto a high frequency wave known as the carrier wave. This process is done in-order to reduce the problem of attenuation (loss of strength of signal) so that the signals can be transmitted to longer distances.

Note:The modulation index is a factor that represents the extent to which an amplitude (for the carrier wave) is changed when compared with the modulating signal. It is also known as the degree of modulation and the modulation index lies between the values $0$ and $1$ usually.