Page 328 - Analog and Digital Filter Design

325
Filters for Phase-Locked Loops

1
F(s) =
l+s.C.R
The damping factor is most important. and is given by:

The lead-lag network will now be considered. This has an additional resistor in
the shunt path, which increases the designer’s options. The lead-lag network
equations are different from those for the RC network, as I will now show. Some
of them are intuitive; others are less so. First the filter’s cutoff frequeiicy is
given by:

wLp = 1/(R1+ R2).C, in rad,’s.

Measuring the signal across the capacitor in this network is the same as in the
CR network, but with R1 and R2 replacing R.

ce, = d(K@. KO. wLp), this is the same as the CR network equation and
is in radls.

Ideally, should have a value between 0.5 and 1.0. A value of < = 0.7071 is
recommended; this is the value used in Butterworth filters for maximally flat
frequency response and has a step response with a slight overshoot. A value of
<= 0.5 has the lowest noise bandwidth (B = 0.5w,,), A value of <= 1.0 has no
step response overshoot.

The transfer function (frequency response if s =jw) of a lead-lag filter is given
by the expression:

R2.C.s-kl
F(s) =
(R1 .C + R2 .C) .s + 1

This gives a frequency response with a 20 &/decade roll-off.

More complex active filters can be used, but more care is needed in their design.
An active lead-lag is the simplest network and is shown in Figure 13.5.