## 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:

Mathematically, it is given as:

ωi = ωc + K

_{f}f(t)
Where ωi = Instantaneous frequency of the modulated wave

ωc = Unmodulated carrier frequency wave

The term

The term

**K**represents the frequency sensitivity of the modulator expressed in_{f}**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.

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

The instantaneous frequency is given by the equation

ωi = ωc + k

_{f}**. x(t)........(1)**Where

**K**

_{f }is proportionality constant and is known as the frequency sensitivity of the modulator. This is expressed in Hz / volt.

**c(t) = A cos (**ωc t + θ0 )........(2)

or

**c(t) = A cos**ϕ .......(3)Here

On differentiating the above equation, we get

dϕdt

**=**ωc .........(4)o

**r**

**Based on above equation, we may write the expression for instantaneous phase angle**ϕ = ∫ωc dt..........(5)

**Where**ϕi = ∫ωi dt............(6)

Putting the value of

ϕi = ∫ [ ωc + k

=ωc t + k∫x(t)dt .

_{f}. x(t)] dt=

_{f}Putting this value of

**c(t) = A cos [**ωc t + k∫ x(t) dt ].Which is the required general expression of FM wave.

_{f}