Question

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

Answer 1

Modulation index is a measure of extent of modulation done on a carrier signal. In Amplitude modulation, it is defined as the ratio of the amplitude of modulating signal to that of the carrier signal.

$m=\frac{A_m}{A_c}$

Its value is kept less than 1 to avoid overmodulation which leads to distortions in the modulated signal and makes it very hard to demodulate and extract the modulating signal.

From the given data,

$m=\frac{A_m}{A_c}=\frac{A_m}{50}$

$0.5=\frac{A_m}{50}$

$A_m=25V$

AM wave is given by,

$x\left(t\right)=A_c(1+m\cos ({\omega }_{m}t\left)\right)\cos \left({\omega }_{c}t\right)$

Hence, amplitude is given by:

$A_x=A_c(1+m\cos ({\omega }_{m}t\left)\right)$

$=50(1+0.5\cos (2\pi \times {10}^{4}t\left)\right)$

Maximum value of AM wave is $x_{max}=50\times 1.5=75V$

Frequency of lower sideband, $f_1=f_c-f_m=1.5\times {10}^6-{10}^4=1.49MHz$

Frequency of upper sideband, $f_1=f_c+f_m=1.5\times {10}^6+{10}^4=1.51MHz$