文档介绍：7-1
Experiment 7: Frequency Modulation and Phase Locked Loops
Frequency Modulation
Normally, we consider a voltage wave form with a fixed frequency of the form
v(t) = V sin(ωct + θ) , (1)
where ωc is the fixed angular frequency and θ is the phase. When we have a voltage wave form
with a variable frequency, this has the form
v(t) = V sin φ(t) (2)
where φ(t) is the total angular displacement at time t. Consistent with this viewpoint, the
instantaneous frequency in radians/sec is
φ
d( (t)) (3)
ωi(t) = 2 π fi(t) =
dt
For example, a constant frequency, ωc, implies
φ(t) = ωct + θ(4)
in which case φ
d( (t)) = ω(5)
ωi = c
dt
In this manner, a frequency modulated wave with sinusoidal modulation has an
instantaneous frequency
fi(t) = fc + ∆f cos (2πfmt) (6)
where fc is the average frequency of the carrier wave, and ∆f is the maximum deviation of the
instantaneous frequency from the average frequency. Notice that ∆f is proportional to the peak
amplitude of the modulating signal and is independent of the modulating frequency, fm. From
equation (3) and equation (6) we see that
d(φ(t))
= ωc + 2 π∆f cos ωmt (7)
dt
2π∆f
φ(t) = ω t + sin ω t + θ(8)
c ω m
m
Neglecting θ for simplicity, we then have the frequency modulated signal from equation (2),
7-2
2π∆f
v(t) = Vsin