Modern Physics Ques 3
Which one of the following best describes the above signal?
(a) $\left[1+9 \sin \left(2 \pi \times 10^4 t\right)\right] \sin \left(2.5 \pi \times 10^5 t\right) \mathrm{V}$
(b) $\left[9+\sin \left(2 \pi \times 10^4 t\right)\right] \sin \left(25 \pi \times 10^5 t\right) \mathrm{V}$
(c) $\left[9+\sin \left(4 \pi \times 10^4 t\right)\right] \sin \left(5 \pi \times 10^5 t\right) \mathrm{V}$
(d) $\left[9+\sin \left(2.5 \pi \times 10^5 t\right)\right] \sin \left(2 \pi \times 10^4 t\right) \mathrm{V}$
Show Answer
Answer:
Correct Answer: 3.( b )
Solution:
20 Equation of an amplitude modulated wave is given by the relation,
$ C_m=\left(A_c+A_m \sin \omega_m t\right) \cdot \sin \omega_c \cdot t $
For the given graph, maximum amplitude,
$ A_c+A_m=10 $ and minimum amplitude, $A_c-A_m=8$ From Eqs. (ii) and (iii), we get
$ \begin{aligned} & A_c=9 \mathrm{~V} \\ & A_m=1 \mathrm{~V} \end{aligned} $
and
$\because$ For angular frequency of message signal and carrier wave, we use a relation
$ \omega_c=\frac{2 \pi}{T_c}=\frac{2 \pi}{8 \times 10^{-6}} $
(as from given graph, $T_c=8 \times 10^{-6} \mathrm{~s}$ )
$ =2.5 \pi \times 10^5 \mathrm{~s}^{-1} $
and
$ \omega_m=\frac{2 \pi}{T_m}=\frac{2 \pi}{100 \times 10^{-6}} $
(as from given graph, $T_m=100 \times 10^{-6} \mathrm{~s}$ )
$ =2 \pi \times 10^4 \mathrm{~s}^{-1} $
When we put values of $A_c, A_m, \omega_c$ and $\omega_m$ in Eq. (i), we get
$ C_m=\left[9+\sin \left(2 \pi \times 10^4 t\right)\right] \sin \left(2.5 \pi \times 10^5 t\right) \mathrm{V} $
BETA
