Q.Explain amplitude modulation. Derive the voltage equation of an AM wave.

Amplitude modulation is the process in which the amplitude of a high-frequency carrier is varied about its unmodulated value in accordance with the instantaneous value of the modulating signal, while its frequency and phase stay constant. Working from the carrier and modulating expressions gives the AM voltage equation as a carrier plus two side bands.

[!ANSWER]

$e_{AM}=E_c(1+m_a\cos\omega_m t)\cos\omega_c t = E_c\cos\omega_c t + \dfrac{m_aE_c}{2}\cos(\omega_c+\omega_m)t + \dfrac{m_aE_c}{2}\cos(\omega_c-\omega_m)t$, i.e. a carrier at $f_c$ and side bands at $f_c\pm f_m$.

[!TLDR]

AM varies the carrier amplitude with the message; its voltage equation $e_{AM}=E_c(1+m_a\cos\omega_m t)\cos\omega_c t$ expands into a carrier plus upper and lower side bands.

What is amplitude modulation: Amplitude modulation (AM) is the process in which the amplitude of the carrier wave is varied in accordance with the instantaneous amplitude of the modulating (message) signal, keeping the carrier frequency and phase unchanged. The shape traced by the varying peaks of the carrier is called the envelope, and it has the same form as the modulating signal.

Derivation of the voltage equation:

Let the carrier voltage be

and the modulating voltage be

where $E_c, E_m$ are peak amplitudes, $\omega_c = 2\pi f_c$, $\omega_m = 2\pi f_m$ and $f_c \gg f_m$.

In AM the instantaneous amplitude of the carrier becomes

With the modulation index $m_a = E_m/E_c$,

The AM wave is this amplitude multiplying the carrier:

Expanding and using $\cos A\cos B = \tfrac{1}{2}[\cos(A+B)+\cos(A-B)]$:

Interpretation: The AM wave has three components:

• the carrier of frequency $f_c$ and amplitude $E_c$,
• the upper side band (USB) of frequency $f_c+f_m$ and amplitude $m_aE_c/2$,
• the lower side band (LSB) of frequency $f_c-f_m$ and amplitude $m_aE_c/2$.

The bandwidth required is the difference between the highest and lowest frequencies present:

The modulation index $m_a$ must not exceed 1; if $m_a>1$ the envelope is distorted (over-modulation).

[!ANSWER]

The AM voltage equation is $e_{AM}=E_c(1+m_a\cos\omega_m t)\cos\omega_c t = E_c\cos\omega_c t + \dfrac{m_aE_c}{2}\cos(\omega_c+\omega_m)t + \dfrac{m_aE_c}{2}\cos(\omega_c-\omega_m)t$; it contains the carrier at $f_c$ and two side bands at $f_c\pm f_m$, giving a bandwidth of $2f_m$.