Q.A sinusoidal carrier voltage $V_C = 80\sin 2\pi\times10^5\,t$ is amplitude modulated by a sinusoidal voltage $v_m = 32\sin 2\pi\times10^3\,t$. Write the equation of the AM wave and draw the output frequency spectrum.

Reading $V_C = 80$ V and $V_m = 32$ V straight from the two sinusoids gives $m_a = V_m/V_C = 0.4$, from which the AM equation and the three-line spectrum (carrier plus two side frequencies of amplitude $m_a V_C/2$) follow. …

[!TLDR]

With $m_a = 32/80 = 0.4$, $v_{AM} = 80(1 + 0.4\sin\omega_m t)\sin\omega_c t$; the spectrum has the carrier (80 V, 100 kHz) and two side frequencies of 16 V at 99 kHz and 101 kHz.

The peak amplitudes and frequencies are read directly from the given sinusoids: $V_C = 80$ V at $f_c = 10^5$ Hz = 100 kHz, and $V_m = 32$ V at $f_m = 10^3$ Hz = 1 kHz. The modulation index is

so the AM wave is

The frequency spectrum is a plot of amplitude versus frequency. The carrier appears as a line of height $V_C = 80$ V at $f_c = 100$ kHz. Each side frequency has amplitude

and they sit symmetrically at …