AN178 Philips, AN178 Datasheet - Page 15

• Manufacturer Part Number: AN178
• Description: Modeling the PLL
• Manufacturer: Philips

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satisfactory tracking. As this value decreases, however, it
attenuates the sum frequency component of the phase comparator
output less. The demodulated signal will appear to have greater
distortion unless this component is filtered out before the distortion is
measured.
NATURAL FREQUENCY AND DAMPING
Circuits and mathematical expressions for the natural frequencies
and dampings are given in Figure 16 for two first-order low-pass
filters. Because of the integrator action of the PLL in converting
frequency to phase, the order of the loop always will be one greater
than the order of the LPF. Hence, both these first-order LPFs
produce a second-order PLL system.
The natural frequency (
can be measured by applying a frequency-modulated signal of the
desired amplitude to the loop. Figure 16 shows that the natural
frequency is a function of K
amplitude. As the modulation frequency (
phase relationship between the modulation and recovered sine
wave will go through 90 at
peak.
1988 Dec
4
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1
Modeling the PLL
IN
.com
Transfer Function
Natural Frequency
n
R 1 C
F(s)
a. Simple
Damping
R
Circuit
1
2K o K
1
K o K d
Figure 16. First-Order Low-Pass Filters
n
1
d
C
1
s 1
OUT
n
) of a loop in its final circuit configuration
d
, which is, in turn, a function of input
m
=
n
1
2
and the output amplitude will
IN
R 1 C
R 2 C
Transfer Function
Natural Frequency
R
n
m
F(s)
1
) is increased, the
a. Lag-Lead
Damping
Circuit
2
n
1
C
R
1
K o K d
( 2
2
1
s (tau 1
2
K o K
OUT
s 2
1
d
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2 )
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Damping is a function of K
and K
amplitude, respectively, damping is highly dependent on the
particular operating condition of the loop. Damping estimates for the
desired operating condition can be made by applying an input signal
which is frequency-modulated within the lock range by a square
wave. The low-pass filter voltage is then monitored on an
oscilloscope which is synchronized to the modulating waveform, as
shown in Figure 17. Figure 18 shows typical waveforms displayed.
The loop damping can be estimated by comparing the number and
magnitude of the overshoots with the graph of Figure 19, which
gives the transient phase error due to a step in input frequency.
An expression for calculating the damping for any underdamped
second-order system ( < 1.0) when the normalized peak overshoot
is known is
Examination of Figure 18 shows that the normalized peak overshoot
of the error voltage is approximately 1.4. Using this value for M
Equation 68 gives a damping of
Another way of estimating damping is to make use of the frequency
response plot measured for the natural frequency (
measurement. For low damping constants, the frequency response
measurement peak will be a strong function of damping. For high
damping constants, the 3dB down point will give the damping.
Figure 19 tabulates some approximate relationships.
Figure 17. Measurement Setup for Display of PLL Transient
M
d
p
are functions of the free-running frequency and input
1
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e
1
d
, K
2
Response
o
, and the lowpass filter. Since K
0.28.
n
Application note
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AN178
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o
p
(68)
in