FM modulation - derivation

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harry456

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The Formula that describes the FM is
Ufm(t)=Uc*cos(2*pi*fc*t+Kfm*integral(0,t,m(t))

with m(t)=sin(2*pi*fm*t)
but the integral of sin(x) is -cos(x). And sin(90°)=1 cos(90°)=0, sin(270°)=-1 cos(270°)=0 ... What I don't get is: At max of m(t) (sin(x)) is the integral(m(t)) =0 that means the frequency is not increased or decreased but the idea behind FM is that at the max of m(t) the frequency should be at its extrema. What do I wrong or what do I not consider?
 

While m(t) is describing the instantaneous frequency, the integral represents the respective phase variation. I don't get how you conclude from phase delta = 0 that the frequency is unchanged? In fact the phase delta zeros coincide with frequency delta extrema and vice versa.

In the usual intuitive view of FM, fc itself is varied. Unfortunately this approach doesn't lead to a simple mathematical representation, so the shown formula does the trick to convert FM to PM, getting a simple additive phase term. The trick doesn't change nature of FM, however.
 

Ufm(t)=Uc*cos(2*pi*fc*t+Kfm*integral(0,t,m(t))

m(t)=sin(2*pi*fm*t)
but the integral of sin(x) is -cos(x) i.e. sin(90°)=1 cos(90°)=0, sin(270°)=-1 cos(270°)=0

does this not lead to Ufm=Uc*cos(2*pi*fc*t+0) for m(t)=max. ? And does this not imply that the frequency is unchanged? (Should be the result be sth like Ufm=Uc*cos(2*pi*fc*t+X) und X should be max dt the max. amplitude of m(t)?
 

To take a different viewpoint, by adding a time-variable phase to the sine function argument (a phase modulation), the instantaneous frequency is surely varied too. That's a simple conclusion from the relation between phase and frequeny modulation.
 

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