Identify the mathematical expression for amplitude modulated wave:

Options

A. `A_c sin[{omega_c +k_1V_m (t)}t+phi] `
B. `A_c sin[{omega_ct +phi +k_2 V_m(t) }`
C. `{A_c+k_2 V_m(t)} sin (omega_ct + phi)`
D. `A_cV_m (t)m sin (omega_ct + phi)`

Answer

C

Explanation

Consider a sinusoidal modulating signal represented by
`m(t) = A_m sin omega_mt " "...(i)`
where, `A_m` = Amplitude of modulating signal
`omega_m` = Angular frequency `= 2pi V_m = phiV_m " "... (ii)`
Also consider a sinusoidal carrier wave represented by
`C(t) = A_c sin omega_ct " "...(ii)`
Thus, modulated wave is given by
`C_m(t) =(A_c + A_m sin omega_mt) sin omega_ct`
`=A_c(1+(A_m)/A_csinomega_mt)sinomega_ct`
Here, `(A_m)/A_c=M`
`implies C_m(t) =(A_c+A_c xxmusin omega_mt)sinomega_ct" "...(iii)`
Now, we know that `A_c xx mu=K` [wave constant]
and `sin omega_mt = V_m` [wave velocity]
Thus, Eq. (iii) becomes
`C_m(t) = (A_c +KxxV_m) sin omega_ct`
Now, consider a change in phase angle by `phi` then `sin omega_ct rarr sin(omega_ct+phi)`
Thus, `C_m(t) = (A_c + KV_m) (sin omega_c+phi)`