# Finding the Power developed by the Nominal Frequency in FM

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?

• Which equation is the one "seen" at the antenna? Oct 12 '14 at 8:35
• The power developed at exactly 100 MHz is zero when modulated with a continuous waveform. Oct 12 '14 at 10:22