Question

What will be the modulation index of the $AM$ wave shown in the figure:

Answer 1

Hint: Learn the definition of AM wave. Study the definition of modulation index and its importance. Use the formula for modulation index for amplitude modulated wave to find the modulation index of the graph given.

Formula used:
The modulation index of a amplitude modulated wave is given by,
$$m_a={\displaystyle \frac{V_{max}-V_{min}}{V_{max}+V_{min}}}$$
Where, $$m_a$$ is the modulation index of the wave $$V_{max}$$ is the maximum amplitude of the of the $AM$ wave and $$V_{min}$$is the minimum amplitude of the $AM$ wave.

Complete step by step answer:
We know that amplitude modulated is one type of modulated wave where the information is modulated into the amplitude of the carrier wave. The modulation factor of a wave decides how well a wave is modulated in the carrier signal. If the value of the modulation index is 1 the wave is said to be fully modulated.

Now, the value of modulation index for amplitude modulated wave is given by,
$$m_a={\displaystyle \frac{V_{max}-V_{min}}{V_{max}+V_{min}}}$$
Now, here we can see from the graph that the maximum voltage amplitude $$V_{max}=13V$$ and the minimum voltage amplitude of the AM wave is $$V_{min}=7V$$.
 Hence, putting the values we will have,
$$m_a={\displaystyle \frac{13-7}{13+7}}$$
$$\Rightarrow m_a={\displaystyle \frac{6}{20}}={\displaystyle \frac{3}{10}}$$
$$\therefore m_a=0.3$$

Hence, the modulation index of the given amplitude modulated wave is $$0.3$$.

Note: The modulation index of any wave is a dimensionless quantity as it is the ratio of the voltages. The percentage modulation is the percentage of the modulation index in this case the modulation percentage is $$0.3\times 100=30\mathrm{\%}$$. It represents that only $$30\mathrm{\%}$$ of the signal wave is modulated in the carrier wave.