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Bandpass Signaling Principles and Circuits Chap. 4
where Θ 0 ( f) = [u 0 (t)] and Θ i ( f ) = [u i (t)]. Of course, the design and analysis techniques
used to evaluate linear feedback control systems, such as Bode plots, which will indicate
phase gain and phase margins, are applicable. In fact, they are extremely useful in describing
the performance of locked PLLs.
The equation for the hold-in range may be obtained by examining the nonlinear behavior
of the PLL. From Eqs. (4–94) and (4–96), the instantaneous frequency deviation of the
VCO from v 0 is
du 0 (t)
= K (4–102)
dt v v 2 (t) = K v K d [sin u e (t)] * f(t)
To obtain the hold-in range, the input frequency is changed very slowly from f 0 . Here the DC
gain of the filter is the controlling parameter, and Eq. (4–102) becomes
¢v = K v K d F(0) sin u e
(4–103)
The maximum and minimum values of ∆v give the hold-in range, and these are obtained
when sin u e = ±1. Thus, the maximum hold-in range (the case with no noise) is
¢f h = 1
(4–104)
2p K vK d F(0)
A typical lock-in characteristic is illustrated in Fig. 4–23. The solid curve shows the
VCO control signal v 2 (t) as the sinusoidal testing signal is swept from a low frequency to a
high frequency (with the free-running frequency of the VCO, f 0 , being within the swept
band). The dashed curve shows the result when sweeping from high to low. The hold-in range
∆f h is related to the DC gain of the PLL as described by Eq. (4–104).
The pull-in range ∆f p is determined primarily by the loop-filter characteristics.
For example, assume that the loop has not acquired lock and that the testing signal is swept slowly
toward f 0 . Then, the PD output, there will be a beat (oscillatory) signal, and its frequency | f in - f 0 |
will vary from a large value to a small value as the test signal frequency sweeps toward f 0 .
v 2 (t)
Direction
of sweep
Pull-in
range
f p
Hold-in
range
f n
Direction
of sweep
0
f 0
f in
Hold-in
range
f h
Hold-in
range
f p
Figure 4–23
PLL VCO control voltage for a swept sinusoidal input signal.