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?
