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Derive frequency modulation
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Answers
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)