What is Frequency Modulation?

Frequency modulation is a system in which the amplitude of the modulated carrier is kept constant, while its frequency and rate of change are varied by the modulating signal. In this type of angle modulation, the instantaneous frequency is varied linearly with a modulating signal f(t) about an unmodulated frequency.

In other words, the instantaneous value of the angular frequency is equal to the frequency of the unmodulated carrier plus a time varying component proportional to f(t).
Mathematically, it is given as:
ωi = ωc + Kf f(t)
Where ωi = Instantaneous frequency of the modulated wave
ωc = Unmodulated carrier frequency wave
The term Kf represents the frequency sensitivity of the modulator expressed in Hz/V.

How Frequency Modulation Works?

By using FM, the amplitude of the carrier is held constant and its frequency is changed by the modulating signal.
This variation in carrier frequency is known as "deviation". It is expressed in kHz per Volt. The deviation of a carrier in frequency is directly proportional to the Audio modulating signal's loudness.

Frequency Modulation Equation

Frequency modulation is that type of angle modulation in which the instantaneous frequency ωi is varied linearly with a message or base band signal x(t) about an unmodulated carrier frequency ωc.

The instantaneous frequency is given by the equation
ωi = ωc + kf. x(t)........(1)Where Kf is proportionality constant and is known as the frequency sensitivity of the modulator. This is expressed in Hz / volt.
Now let the expression for unmodulated carrier signal be
c(t) = A cos (ωc t + θ0)........(2)
or
c(t) = A cos ϕ.......(3)Here ϕ is the total phase angle of the unmodulated carrier and is given by= ωc t + θ0.

On differentiating the above equation, we get
dϕdt = ωc.........(4) or ϕ = ωc dt..........(5)Based on above equation, we may write the expression for instantaneous phase angle θi as
ϕi = ωi dt............(6) Where ωi = instantaneous frequency of frequency modulated wave.

Putting the value of ωi in equation (6), we get
ϕi = [ ωc + kf. x(t)] dt
= ωc t + kf x(t)dt.

Putting this value of ϕi in equation (3), the expression for frequency modulated wave will be
c(t) = A cos [ ωc t + kf x(t) dt ].Which is the required general expression of FM wave.