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Question

The sinusoidal signal \$f(t) = \cos(2 \pi f_m t)\$ is applied to the input of an FM system. The corresponding modulation signal output for \$f_m = 1kHz\$, is

$$Q(t)=100 \cos(2 \pi 100 \times 10^6 t + 4 \sin(w \pi 1000t)) \space > V$$

across 50 ohm resistive load.

What is the power developed at 100MHz?

My Work

\$Q(t) = 100\cos(2 \pi 100 \times 10^6 t + 4 \sin(w \pi 1000t))\$
\$\space\space\space\space\space\space\space\space = 100 \cos(2 \pi 100 \times 10^6 t) \cos(4 \sin(2 \pi 1000 t)) - 100 \sin (2 \pi 100 \times 10^6 t) \sin(4 sin(2 \pi 1000 t))\$

Where
\$\space\space\space\space\space\space\space\space\cos(4 \sin(2 \pi 1000 t) \approx 1\$ and;
\$\space\space\space\space\space\space\space\space\sin(4 sin(2 \pi 1000 t)) \approx 4 \sin(2 \pi 1000 t)\$
(These relationships are obtained from my lecture note)

Therefore,
\$\space\space\space\space\space\space\space\space Q(t) = 100 \cos(2 \pi 100 \times 10^6 t) - 100 \sin (2 \pi 100 \times 10^6 t) \times 4\sin (2 \pi 1000 t)\$

Using equation \$P=\frac{V^2}{2R}\$, \$P\space =\space \frac{100^2}{2\cdot 50}=100\space W\$.

There are two things containing \$100*10^6\$ Hz, I do not know whether it is correct... Can you explain more about this question to me?

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  • \$\begingroup\$ Which equation is the one "seen" at the antenna? \$\endgroup\$
    – Dejvid_no1
    Oct 12 '14 at 8:35
  • \$\begingroup\$ The power developed at exactly 100 MHz is zero when modulated with a continuous waveform. \$\endgroup\$
    – Andy aka
    Oct 12 '14 at 10:22
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Power developed at 100 MHz can be easily calculated by using Bessel function.

By Bessel function, J0(4) = -0.4

P= 1/2R * Ac^2 * J0(4)^2 = 800/R W

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