Question

Identify the mathematical expression for amplitude modulated wave:

Answer 1

Correct option is C. $$\{ A_c + k_2 v_m (t)\} \sin (\omega_c t + \phi)$$
Consider a signal to be modulated is $$m(t) = A_m \sin \omega_mt$$ where $$A_m$$ is amplitude of modulated signal
$$A_m = A_c + m(t)$$
$$A = A_c + A _m \sin \omega_m t$$
$$A_m = A_c \left [ 1 + \frac{A_m}{A_c} \sin \omega_m t \right ] \left [ \because \mu = \frac{A_m}{A_c} \right ]$$
$$\omega_m = 2 \pi v_m \phi v_m (V_m =$$ frequency of signal)
$$C(t) = A_c \sin \omega_e t$$
$$C_m(t) = C(t) + m(t)$$
$$C_m(t) = A_m \sin \omega_c t$$
$$C_m(t) = A_c [1 + \mu \sin \omega_m t] \sin \omega_c t$$
$$= {A_c + \mu A_c \sin \omega_m t} \sin (\omega_c t + \phi) (\phi =$$ phase diff.)
Let $$\mu A_c = k_2$$
Sin $$\omega_m t = v_m(t)$$
$$\therefore C_m(t) = \{ A_C + K_2 v_m (t)\} \sin \phi $$)
So verifies answer (c)