Q100 / 120 · VITEEE 2007 · Physics
The antenna current of an AM transmitter is $8 \mathrm{~A}$ when only the carrier is sent, but it increases to $8.93 \mathrm{~A}$ when the carrier is modulated by a single sine wave. Find the percentage modulation.
- A$60.1 \%$
- Correct answerB$70.1 \%$
- C$80.1 \%$
- D$50.1 \%$
Solution
The rms value of carrier current is $\mathrm{I}_{\mathrm{c}}=8 \mathrm{~A}$. The rms value of modulated current
$=I_{c}+I_{m}=8.93 \mathrm{~A}=\mathrm{I}_{\mathrm{t}}$
Percentage modulation $=\mathrm{m}_{\mathrm{a}} \times 100$
The current relation in AM wave is
$\frac{\mathrm{I}_{\mathrm{t}}^{2}}{\mathrm{I}_{\mathrm{c}}^{2}}=1+\frac{\mathrm{m}_{\mathrm{a}}^{2}}{2} \Rightarrow \mathrm{m}_{\mathrm{a}}=\sqrt{\left(\frac{\mathrm{I}_{\mathrm{t}}^{2}}{\left.\mathrm{I}_{\mathrm{c}}^{2}-1\right) 2}\right.}$
$\Rightarrow$ Modulation index, $\mathrm{m}_{\mathrm{a}}=\sqrt{\left(\frac{(8.93)^{2}}{8^{2}}-1\right) 2}$
$\begin{aligned}=\sqrt{\left(\frac{79.7}{64}-1\right) 2} &=\sqrt{(1.24-1) 2} \\ &=0.701 \\ \text { Percentage modulation } &=\mathrm{m}_{\mathrm{a}} \times 100=70.1 \% \end{aligned}$
Hence, the correct answer is option: 'B'.