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

4.8 Phase-locked Loop and Some Applications 177
We can use small-error analysis, to show that a first-order loop cannot track an incoming
signal whose instantaneous frequency varies linearly with time. Moreover, such a signal can be
tracked within a constant phase ( constant phase error) by using a second-order loop [Eq. ( 4.37)],
and it can be tracked with zero phase error by using a third-order loop. 7
It must be remembered that the preceding analysis assumes a linear model, which is valid
only when 0 e (t) « n /2. This means the frequencies wo and We must be very close for this
analysis to be valid. For a general case, one must use the nonlinear model in Fig. 4.26b. For
such an analysis, the reader is referred to Viterbi, 7 Gardner, 8 or Lindsey. 9
First-Order Loop Analysis
Here we shall use the nonlinear model in Fig. 4.26b, but for the simple case of H (s) = 1. For
this case h(t) = 8(t),* and Eq. (4.33) gives
Because 0 e = 0i - 0 o ,
B 0 (t) = AK sin 0 e (t)
(4.39)
Let us here consider the problem of frequency and phase acquisition. Let the incoming
signal be A sin (wot + <po) and let the VCO have a quiescent frequency We , Hence,
and
B e = (wo - W e ) -AK sin 0 e (t) (4.40)
For a better understanding of PLL behavior, we use Eq. ( 4.40) to sketch B e vs. 0 e , Equation
( 4.40) shows that B e is a vertically shifted sinusoid, as shown in Fig. 4.28. To satisfy Eq. ( 4.40),
the loop operation must stay along the sinusoidal trajectory shown in Fig. 4.28. When B e = 0,
the system is in equilibrium, because at these points, 0 e stops varying with time. Thus 0 e =
01 , 02, 03, and 04 are all equilibrium points.
If the initial phase error 0 e (O) = 0 e o (Fig. 4.28), then B e corresponding to this value of 0 e
is negative. Hence, the phase error will start decreasing along the sinusoidal trajectory until it
Fi g ure 4.28
Trajectory of a
first-order PLL.
e,t
* Actually h(t) = 2B sine (2n Bt), where B is the bandwidth of the loop filter. This is a low-pass, narrow band filter,
which suppresses the high-frequency signal centered at 2wc, This makes H (s) = 1 over a low-pass narrow band of B
Hz.