# (a)    If $A$ and $B$ represent the maximum and minimum amplitudes of an amplitude modulated wave, write the expression for the modulation index in terms of $A$ & $B$.(b)    A message signal of frequency $\inline 20\; kHz$ and peak voltage $\inline 10\; V$ is used to modulate a carrier of frequency $\inline 2\; MHz$ and peak voltage of $\inline 15\; V$. Calculate the modulation index. Why the modulation index is generally kept less than one?

(a)  We have given that:-

Maximum amplitude,$A_{max}=A$

Minimum Amplitude,$A_{min}=B$

Let, the amplitude of the carrier wave be $A_{c}$ and

the amplitude of the message signal be $A_{m}$

Now, we know that,

$A_{c}+A_{m}=A_{max}=A$ _____(1)

$A_{c}-A_{m}=A_{min}=B$ ______(2)

On adding(1) and (2), we have

$2\; A_{c}=A+B$

$A_{c}=\frac{A+B}{2}$

and, On sunstracting (2) from (1), we have

$2\; A_{m}=A-B$

$A_{m}=\frac{A-B}{2}$

Hence, we know, the modulation index in terms of

$A$ and $B$  can be given as ;

$\mu =\frac{A_{m}}{A_{c}}$

$\mu =\frac{A-B}{A+B}$

(b)     We know, the modulation index is given by :-

$\mu =\frac{A_{m}}{A_{c}}$

Hence, $A_{m}$ is given as $10\; V$ and

$A_{c}$ is given as $15\; V$  So,

$\mu =\frac{10\; V}{15\; V}$

$\mu =\frac{2}{3}$

To avoid the Distortion of the signal, the modulation index is generally kept less than one.

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