Question

Define modulation index. Why is the value kept, in practice, less than one? A carrier wave frequency 1.5MHz and amplitude 50V is modulated by a sinusoidal wave of frequency 10 kHz producing 50% amplitude modulation. Calculate the amplitude of the AM wave and frequencies of the side bands produced.

Answer 1

Hint: Modulation index is ratio of the amplitude of the modulated wave to the amplitude of the carrier wave which should be less than one so the amplitude of the modulated wave remains less than that of the carrier wave.
We can solve the given numerical by keeping in mind the following formulas
Modulation index \[\mu =\dfrac{{{A}_{m}}}{{{A}_{c}}}\]
Frequency of lower side band \[{{f}_{1}}={{f}_{c}}-{{f}_{m}}\]
Frequency of the upper side band \[{{f}_{1}}={{f}_{c}}+{{f}_{m}}\]

Complete step-by-step answer:
Modulation index is defined as a measure of the extent of modulation done on a carrier signal wave. In Amplitude modulation, it can be defined as the ratio of the amplitude of a modulating signal to that of the amplitude of the carrier signal.
\[\mu =\dfrac{{{A}_{m}}}{{{A}_{c}}}\]
Its value is generally kept less than 1 for practical purposes to avoid over modulation which leads to distortions in the modulated signal and then makes it very hard to demodulate and extract the modulating signal.

From the given data,
Frequency of the carrier wave,
\[{{f}_{c}}=1.5MHz=1500kHz\]
Frequency of the sinusoidal or modulated wave,
\[{{f}_{m}}=10kHz\]
Amplitude of the carrier wave,
\[{{A}_{c}}=50V\]
Modulation index \[\mu =50\%=\dfrac{1}{2}\]
So, now using this we can calculate Amplitude of the modulated wave,
\[\mu =\dfrac{{{A}_{m}}}{{{A}_{c}}}=\dfrac{{{A}_{m}}}{50}\]​​
\[\dfrac{{{A}_{m}}}{50}=0.5\]
\[{{A}_{m}}=25V\]

Amplitude Modulated wave is given by,
\[x\left( t \right)={{A}_{c}}\left( 1+\mu \cos \left( {{\omega }_{m}}t \right)\cos \left( {{\omega }_{c}}t \right) \right)\]
Hence, amplitude will be given by:
\[{{A}_{x}}={{A}_{c}}\left( 1+\mu \cos \left( {{\omega }_{m}}t \right) \right)\]
\[{{A}_{x}}=50\left( 1+0.5\cos \left( 2\pi \times {{10}^{4}}t \right) \right)\]
Maximum value of AM wave is \[{{x}_{\max }}=50\times 1.5=75V\]
Frequency of lower sideband is given by, \[{{f}_{1}}={{f}_{c}}-{{f}_{m}}=1500-10=1490kHz\]
Frequency of upper sideband is given by, \[{{f}_{1}}={{f}_{c}}+{{f}_{m}}=1500+10=1510kHz\]

Note: Demodulation is the process of extracting the original signal that bears information from a carrier wave. A demodulator is an electronic circuit device that is used to recover the information content from the modulated carrier wave. There are many types of modulation so there are many types of demodulators which are used to demodulate the modulated wave.