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

176 AMPLITUDE MODULATIONS AND DEMODULATIONS
to generate a local signal of frequency cvo and phase* <po. Assuming the quiescent frequency
of the VCO to be ev e , the incoming signal can be expressed as A sin [w e t + 0;(t)], where
0;(t) = (cvo - cv c )t + <po
and
G; (s) = cvo - We + <po
s 2 s
Consider the special case of H (s) = 1. Substituting this equation into Eq. ( 4.35),
Hence,
G e (s) = __ s _ cvo - W e
[
<po]
+
s + AK s 2 s
(cv 0 - cv e )/AK (WO - W e )/AK <po
=---- ----- +--
s s + AK s + AK
(4.36a)
Observe that
lim 0 e (t) =
t-+oo
WO - CV c
AK
(4.36b)
Hence, after the transient dies (in about 4/AK seconds), the phase error maintains a constant
value of (cvo - cv c )/AK. This means the PLL frequency eventually equals the incoming
frequency cvo. There is, however, a constant phase error. The PLL output is
B cos [ wot + <po -
cvo -cvc
J
AK
For a second-order PLL using
s + a
H(s) = -- s
s
G e (s) = G; (s)
s + AKH (s)
(4.37a)
(4.37b)
the final value theorem directly yields, 6
s 2 [ cvo - W e <po - ----- --- + - ]
s
2 + AK(s + a) s
2
s
lim (Je(t) = lim sG e (s) = 0
t--+oo s--+0
(4.38)
In this case, the PLL eventually acquires both the frequency and the phase of the incoming
signal.
* With a difference re /2.