Page 42 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL BUILDING BLOCKS Ronald E. Best 29
DC component that is roughly proportional to the phase error θ ; the remaining terms are AC
e
components having frequencies of 2 ω , 4 ω …
1
1
Because these higher frequencies are unwanted signals, they are filtered out by the loop
filter. Because the loop filter must pass the lower frequencies and suppress the higher, it must
be a low-pass filter. In most PLL designs, a first-order low-pass filter is used. Three different
types of loop filters are mostly used in PLL circuits: the passive lead-lag filter, the active lead-
lag filter, and the active PI filter. These will be discussed in Secs. 2.5.1 through 2.5.3.
Every loop filter is driven by a phase detector. We have seen there are phase detectors with
voltage output and those with current output. The loop filters used in combination with a
phase detector having a voltage output differs slightly from a loop filter designed to be used
with a charge pump phase detector. Hence, all the filters discussed in the following sections
will be shown with two versions.

Type 1: Passive lead-lag filters

Figure 2.17 shows first-order passive lead-lag filters having one pole and one zero. Figure
2.17a is the version used in combination with a voltage output phase detector. Its transfer
function F(s) is given by

(2.29a)

where τ = R C and τ = R C. F(s) is the ratio of the Laplace transform of the averaged filter
2
1
1
2
output signal and of the Laplace transform of the averaged phase detector output voltage
signal . (A note on terminology: a lead-lag [also called lag-lead] filter combines a phase-
leading with a phase-lagging network. The phase-leading action comes from the numerator [in
other words, from the zero] in the transfer function in Eq.(2.29a), whereas the denominator
[that is, the pole] produces the phase lag. All filters used as loop filters are lead-lag filters).
When a phase detector with current output drives the loop filter resistor, R becomes
1
useless, so it is sufficient to use a loop filter as shown in Fig. 2.17b. For this loop filter, the
transfer function is

(2.29b)

Here, F(s) is the ratio of the Laplace transforms of output voltage u and input current i ; thus,
f d
it has the dimension Ohm.
The amplitude response of the passive lead-lag loop filters is shown in Fig. 2.18. Figure
2.18a is the Bode diagram of the voltage-driven filter, while Fig. 2.18b is the Bode diagram of
the current-driven filter. As we will see in Sec. 3.4, the zero of this filter is crucial because it
has a strong influence on the damping factor ζ of the PLL system.

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