AN178 Philips, AN178 Datasheet - Page 3

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

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frequency, this beat note would appear at the filter output with
negligible anenuation.
Now suppose that the feedback loop is closed by connecting the
low-pass filter output to the VCO control terminal. The VCO
frequency will be modulated by the beat note. When this happens,
process, the VCO frequency moves closer to
and the output of the phase comparator becomes a slowly varying
function of time. Similarly, if the VCO is modulated away from
becomes a rapidly varying function of time. Under this condition the
beat note waveform no longer looks sinusoidal; it looks like a series
of aperiodic cusps, depicted schematically in Figure 2a. Because of
its asymmetry, the beat note waveform contains a finite DC
component that pushes the average value of the VCO toward
lock is established. When the system is in lock,
and only a steady-state DC error voltage remains.
Figure 2b displays an oscillogram of the loop error voltage V
an actual PLL system during the capture process. Note that as lock
is approached,
becomes less, and the amplitude of the beat note increases.
The total time taken by the PLL to establish lock is called the pull-in
time . Pull-in time depends on the initial frequency and phase
differences between the two signals as well as on the overall loop
gain and the low-pass filter bandwidth. Under certain conditions, the
pull-in time may be shorter than the period of the beat note and the
loop can lock without an oscillatory error transient.
A specific case to illustrate this is shown in Figure 3. The 565 PLL
is shown acquiring lock within the first cycle of the input signal. The
PLL was able to capture in this short time because it was operated
as a first-order loop (no low-pass filter) and the input tone-burst
frequency was within its lock and capture range.
1988 Dec
4
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Modeling the PLL
itself will become a function of time. If, during this modulation
Figure 2. Asynchronous Error Beat Frequency During the
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, (i.e., decreasing
,
d
a. VCO Control Voltage Variation During Capture Transient
dt
e
increases and the error voltage
a. Oscillogram Showing a Capture Process
is reduced, the low-pass filter anenuation
Capture Transient
), then
d
dt
e
decreases
is equal to zero
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(t) in
I
and
3
EFFECT OF THE LOW-PASS FILTER
In the operation of the loop, the low-pass filter serves a dual
function.
First, by anenuating the high frequency error components at the
output of the phase comparator, it enhances the
interference-rejection characteristics; second, it provides a
short-term memory for the PLL and ensures a rapid recapture of the
signal if the system is thrown out of lock due to a noise transient.
Decreasing the low-pass filter bandwidth has the following effects on
system performance (Long Time Constant):
a. The capture process becomes slower, and the pull-in time
b. The capture range decreases.
c. Interference-rejection properties of the PLL improve since the
d. The transient response of the loop (the response of the PLL to
The last effect also produces a practical limitation on the low-pass
loop filter bandwidth and roll-off characteristics from a stability
standpoint. These points will be explained further in the following
analysis.
MATHEMATICALLY DEFINING PLL OPERATION
As mentioned previously, the phase comparator is basically an
analog multiplier that forms the product of an RF input signal, V
and the output signal, v
assume that the two signals to be multiplied can be described by
where
phase error) characteristics of interest. The product of these two
signals is an output voltage given by
where K
amplitude of v
signal V
locked loop (
increases.
error voltage caused by an interfering frequency is attenuated
further by the low-pass filter.
sudden changes of the input frequency within the capture range)
becomes underdamped.
Figure 3. Exhibited by First-Order Fast Capture Transient
V
V
V
i
o
e
(t)
(t)
(t)
l
I
1
. The two cases of an unlocked loop (
,
is an appropriate dimensional constant. Note that the
O
, and
V
V
K
e
I
I
(t) is directly proportional to the amplitude of the input
O
1
=
sin
V
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sin (
I
V
O
e
O
) are now considered separately.
(sin
are the frequency and phase difference (or
I
t
o
O
(t), from the VCO. Refer to Figure 1 and
t
I
t) [sine(
e
)
O
t
e
)]
I
O
Application note
AN178
) and of a
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i
(t),
(3)
(4)
(5)