Page 99 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 67
is plotted against time for two cases. In Fig. 3.10a, the peak frequency deviation is less than
the offset Δω between the reference frequency ω and the down-scaled frequency ω ′.
1
0
Hence, a lock-in process will not take place, at least not instantaneously.
Figure 3.10b shows a special case, however, where the peak frequency deviation is just as
large as the frequency offset Δω. The frequency ω ′ of the VCO output signal develops as
2
shown by the solid line. When the frequency deviation is at its largest, ω ′ exactly meets the
2
value of the reference frequency ω . Consequently, the PLL locks within one single-beat note
1
between the reference and output frequencies. This corresponds exactly to the lock-in process
described in Sec. 3.8 by means of the mechanical analogy. Under this condition, the difference
Δω is identical with the lock range Δω , and we have
L

This is a nonlinear equation for Δω . Its solution becomes very simple, however, if an
L
approximation is introduced for |F(Δω )|. It follows from practical considerations that the lock
L
range is always much greater than the corner frequencies 1/τ and 1/τ of the loop filter. For
1
2
the filter gain |F(Δω )|, we can therefore make the following approximations:
L

Moreover, τ is normally much smaller than τ , so we can use the simplified relationship
2
1

Making use of the substitutions [Eqs. (3.13) to (3.15)] and assuming high-gain loops, we get

(3.61)

for all types of loop filters.
Knowing the approximate size of the lock range, we are certainly interested in having some
indication of the lock-in time. When the PLL locks in quickly, the signals u and u perform
f
d
(for ζ < 1) a damped oscillation, whose angular frequency is approximately ω . As Fig. 3.6
n