GATE 2017 EC (EC2) – Question 61
A modulating signal $x(t)=5\sin[4\pi10^3t-10\pi\cos(2\pi10^3t)]$ V is fed to a phase modulator with phase-deviation constant $k_p=5$ rad/V. The carrier frequency is 20 kHz. The instantaneous frequency (in kHz) at t=0.5 ms is ________.
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Correct answer: 69.9 to 70.1
Explanation
For phase modulation, total phase is $2\pi f_ct+k_px(t)$, so $f_i=f_c+[k_p/(2\pi)]x'(t)$.
Write $x(t)=5\sin\phi(t)$. At t=0.5 ms, $\phi=2\pi+10\pi=12\pi$ and $\cos\phi=1$. Also $\phi'=4\pi10^3+20\pi^2 10^3\sin(2\pi10^3t)$; the sine term is zero at this instant.
Thus $x'=5(4\pi10^3)=20000\pi$ V/s. The frequency increment is $5(20000\pi)/(2\pi)=50000$ Hz. Add the 20 kHz carrier to obtain **70 kHz**.