An FM modulator with frequency sensitivity k f = 10 Hz/V has the
An FM modulator with frequency sensitivity kf = 10 Hz/V has the message signal shown in the figure below.
A. 10π t<sup style="">2</sup>
B. 20π t<sup style="">2</sup>
C. 20πt
D. None of these
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Right Answer is: A
SOLUTION
We have FM signal as:
\(s\left( t \right) = {A_c}\cos \left[ {{\omega _c}t + 2\pi \times 10\mathop \smallint \nolimits_0^t m\left( \tau \right)d\tau } \right]\)
For the time duration 0 ≤ t < 1, we have the message signal as:
m(t) = t
Therefore, the phase deviation for the duration is given by:
\(\theta \left( t \right) = 2\pi \times 10\mathop \smallint \nolimits_0^t m\left( \tau \right)d\tau \)
\(= 2\pi \times 10\mathop \smallint \nolimits_0^t \tau \;d\tau \)
\( = 2\pi \times 10\left[ {\frac{{{\tau ^2}}}{2}} \right]_0^t\)
= 10πt2