Q.Derive an expression for the instantaneous value of a FM wave.

Frequency modulation (FM) is the process by which the frequency of the carrier is varied in accordance with the instantaneous amplitude of the modulating signal, while the carrier's amplitude stays constant. When the message is zero the carrier stays at its rest (centre) frequency $f_c$; a rising message raises the carrier frequency and a falling message lowers it, the extremes occurring at the message peaks.

FM and phase modulation both vary the total phase angle $\phi = (\omega_c t + \theta)$ of the carrier, so together they are called angle modulation. In phase modulation (PM) it is the carrier's phase that is varied in proportion to the message, with amplitude and frequency held constant.

Starting from a message $v_m = V_m\cos\omega_m t$ and carrier $v_c = V_c\sin(\omega_c t + \theta)$, the instantaneous angular frequency during FM is $\omega = \omega_c + K V_m\cos\omega_m t$, where $K$ is the modulator's frequency-sensitivity constant. Integrating the phase leads to the standard FM voltage equation

$v_{FM} = V_c \sin\!\left(\omega_c t + m_f \sin\omega_m t\right)$ …