B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

174 AMPLITUDE MODULATIONS AND DEMODULATIONS
Figure 4.26
Phase-locked
loop and its
equivalent
circuit.
A sin fw c t + om]
Loop
filter
H(s)
Voltagecontrolled
oscillator
eo(t)
(a)
0;(1) sin( ) AKH(s)
8 0 = ce 0 (t)
0/t)
r 0
(b)
Basic PLL Operation
The operation of the PLL is similar to that of a feedback system (Fig. 4.26a). In a typical
feedback system, the feedback signal tends to follow the input signal. If the feedback signal
is not equal to the input signal, the difference (known as the error) will change the feedback
signal until it is close to the input signal. A PLL operates on a similar principle, except that the
quantity fed back and compared is not the amplitude, but the phase. The VCO adjusts its own
frequency such that its frequency and phase can track those of the input signal. At this point,
the two signals are in synchronism (except for a possible difference of a constant phase).
The voltage-controlled oscillator (VCO) is an oscillator whose frequency can be linearly
controlled by an input voltage. If a VCO input voltage is e 0 (t), its output is a sinusoid with
instantaneous frequency given by
w(t) = W e + ce 0 (t) (4.30)
where c is a constant of the VCO and W e is the free-running frequency of the VCO [ when
e 0 (t) = O]. The multiplier output is further low-pass-filtered by the loop filter and then applied
to the input of the VCO. This voltage changes the frequency of the oscillator and keeps the
loop locked by forcing the VCO output to track the phase (and hence the frequency) of the
input sinusoid.
If the VCO output is B cos [w e t+ 0 0 (t)], then its instantaneous frequency is W e + 0 0 (t).
Therefore,
(4.31)
Note that c and B are constant parameters of the PLL.
Let the incoming signal (input to the PLL) be A sin [w e t + 0i (t)]. If the incoming signal
happens to be A sin [w 0 t + i,{!(t)], it can still be expressed as A sin [w e t+ 0; (t)], where 0;(t) =
(w 0 - w c )t + i,/J(t). Hence, the analysis that follows is general and not restricted to equal
frequencies of the incoming signal and the free-running VCO signal.
The multiplier output is