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Derive the expression for frequency modulated wave.

Subject : Principle of Communication Engineering

Topic : Angle modulation and demodulation

Difficulty : High

pce(44) • 12k  views
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• If the frequency of carrier signal is varied in accordance with the instantaneous amplitude of the modulating signal, then it is called as frequency modulation.
• In FM the amplitude of carrier remains constant. The variation in carrier frequency from the umodulated carrier is called frequency deviation (δ).
• FM is non-linear process & information contain in frequency deviation.

Mathematical Expression of FM:

• Instantaneous frequency of frequency modulated wave is;

Wi=Wc+δ

• The frequency deviation above or below Wc depends on the instantaneous amplitude of modulating signal i.e

δ=kVm cos Wmt

where k= frequency deviation sensitivity of FM

• Again, instantaneous frequency is:

Wi=Wc+δ

Wi=Wc+kVm cos Wmt ---------------------(i)

• The frequency modulated signal is represented mathematically as

Vfm=Vc sin θ ------------------------(ii)

where θ instantaneous phase & it can be determine from instantaneous frequency as

Wi = dθ /dt

Wi dt = dθ

∫dθ= ∫Wi dt

Substituting the value of Wi from equation (i) in above equation;

θ= ∫ (Wc+kVm cos Wmt ) dt

θ= Wct+ (kVm/Wm) sin Wmt

θ= Wct+Mf sin Wmt ----------------------------------------(iii)

where Mf (modulation Index)= (kVm/Wm)

Substituting the value of θ from equation (iii) in equation (ii);

Vfm=Vc sin (Wct+Mf sin Wmt)

Above is the equation of frequency modulation.