Question

When only the carrier is transmitted, antenna current observed is $8\,A$ , when it is modulated with $500\,Hz$ sine wave, antenna current becomes $9.6\,A$ . The percentage of modulation is
A. $80\%$
B. $20\%$
C. $93.8\%$
D. $83.76\%$

Answer 1

The modulation index or the modulation depth describes how much the modulated variable of the carrier signal compared to the unmodulated level. The percentage modulation is defined as the original frequency deviation that is produced by the modulating signal to that of the maximum frequency that is allowable. In this problem we will calculate modulation index by using the formula then we will convert into percentage by multiplying it with $100$

Formula used:
$\dfrac{{{I_T}}}{{{I_C}}} = \sqrt {\left( {1 + \dfrac{{{m^2}}}{2}} \right)}$
Where, ${I_T} =$ Antenna current before modulation, ${I_C} =$Antenna current after modulation and $m =$Modulation index.

Complete step by step answer:
Given:
Antenna current before modulation $\left( {{I_T}} \right) = 8A$
Antenna current after modulation $\left( {{I_C}} \right) = 9.6A$
Percentage of modulation $\left( {m\% } \right) = ?$
We know that,
$\dfrac{{{I_T}}}{{{I_C}}} = \sqrt {\left( {1 + \dfrac{{{m^2}}}{2}} \right)}$
Squaring on both sides, we get
${\left( {\dfrac{{{I_T}}}{{{I_C}}}} \right)^2} = \left( {1 + \dfrac{{{m^2}}}{2}} \right)$
On simplifying the above equation
${m^2} = 2\left( {{{\left( {\dfrac{{{I_T}}}{{{I_C}}}} \right)}^2} - 1} \right)$
Substituting the given data in the above equation, we get
${m^2} = 2\left( {{{1.2}^2} - 1} \right) \\\ \Rightarrow {m^2} = 0.88$
Taking square root on both, we get
$m = 0.938$
Expression in terms of percentage
$m = 0.938 \times 100$
$\therefore m = 93.8\%$

Hence, option C is correct.

Note: A carrier signal is transmitted in the form of electromagnetic waves or electromagnetic pulse whose base frequency will be steady on which the information will be imposed by means of increasing the strength of the signal, by varying the base frequency or by varying the wave phase or by other means. This variation is called modulation. Therefore modulation is the process of changing one or more properties of a periodic waveform; this periodic waveform is known as the carrier signal, with an individual signal called a modulation signal which contains information that is to be transmitted.