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A $100V$ carrier wave is made to vary between $160V$ and $40V$ by a modulation signal. What is the modulation index?

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Last updated date: 17th Jul 2024
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Answer
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Hint: In order to solve this question you have to remember the definition of modulation index and the formula for finding the modulation index. Recall all the concepts related to the modulation, types of modulation, characteristics of the modulation.

Formula used:
$\mu = \dfrac{{{A_m}}}{{{A_c}}}$
Where, ${A_m}$ is the amplitude of the modulation signal
${A_c}$ is the amplitude of the carrier signal
And, $\mu $ is the modulation index

Complete step by step solution:
Here, in the question, we have given the amplitude of the carrier signal,
${A_c} = 100V$
Also given the maximum and minimum amplitude of the modulated wave,
${A_{\max }} = 160V$ and ${A_{\min }} = 40V$
We also know that,
$ \Rightarrow {A_{\max }} = {A_c} + {A_m}$
$ \Rightarrow {A_{\min }} = {A_c} - {A_m}$
Where ${A_m}$ is the amplitude of the modulation signal
${A_c}$ is the amplitude of the carrier signal.
On putting the value of ${A_{\max }}$ in the above equation for the maximum amplitude of the modulated wave, we get,
$ \Rightarrow 160V = {A_c} + {A_m}$
On putting the value of ${A_c}$ in the above equation,
$ \Rightarrow 160V = 100 + {A_m}$
On further solving, we get the value of the amplitude of the modulation signal,
${A_m} = 60V$
Now we have the formula for calculating the modulation index, given by,
modulation index, $\mu = \dfrac{{{A_m}}}{{{A_c}}}$
Where ${A_m}$ is the amplitude of the modulation signal
${A_c}$ is the amplitude of the carrier signal
On putting the values of ${A_m}$ and the ${A_c}$ in the above formula, we get
$\mu = \dfrac{{60V}}{{100V}}$
On further solving we get the value of the modulation index,
$\mu = 0.6$

Thus the value of the modulation index is $0.6$.

Additional Information: There is one more formula for finding the modulation index using the maximum and minimum amplitude of the modulated wave. The formula is given by, $\mu = \dfrac{{{A_{\max }} - {A_{\min }}}}{{{A_{\max }} + {A_{\min }}}}$.

Note: The modulation index is also known as modulation depth. If the modulating index is multiplied by 100 then it is converted into the percentage of modulation. If the value of the modulating index is 1, then it is the example of a perfect modulating, and its percentage of modulation is 100%.