AN178 Philips, AN178 Datasheet - Page 2

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

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Philips Semiconductors
INTRODUCTION
The phase-locked loop is a feedback system comprised of a phase
comparator, a low-pass filter and an error amplifier in the forward
signal path and a voltage-controlled oscillator (VCO) in the feedback
path. The block diagram of a basic PLL system is shown in Figure
1. Perhaps the single most important point to realize when
designing with the PLL is that it is a feedback system and, hence, is
characterized mathematically by the same equations that apply to
other, more conventional feedback systems. However, the
parameters in the equations are somewhat different since the
feedback error signal in the phase locked system is a phase rather
than a current or voltage signal, as is usually the case in
conventional feedback systems.
PHASE-LOCKED LOOP OPERATION
The basic principle of the PLL operation can be briefly explained as
follows:
With no signal input applied to the system, the VCO control voltage
V
the equivalent radian frequency
free-running frequency. When an input signal is applied to the
system, the phase comparator compares the phase and the
frequency of the input with the VCO frequency and generates an
error voltage V
diHerence between the two signals. This error voltage is then
filtered, amplified, and applied to the control terminal of the VCO. In
this manner, the control voltage V
vary in a direction that reduces the frequency difference between
and the input signal. If the input frequency
or lock with the incoming signal. Once in lock, the VCO frequency is
identical to the input signal except for a finite phase difference.
This net phase difference of
is necessary to generate the corrective error voltage V
VCO frequency from its free-running value to the input signal
frequency
ability of the system also allows the PLL to track the frequency
changes of the input signal once it is locked. The range of
frequencies over which the PLL can maintain lock with an input
signal is defined as the “lock range” of the system. The band of
frequencies over which the PLL can acquire lock with an incoming
signal is known as the “capture range” of the system and is never
greater than the lock range.
Another means of describing the operation of the PLL is to observe
that the phase comparator is in actuality a multiplier circuit that
mixes the input signal with the VCO signal. This mix produces the
sum and difference frequencies
the loop is in lock, the VCO duplicates the input frequency so that
4
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d
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SIGNAL
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INPUT
(t) is equal to zero. The VCO operates at a set frequency, f
Modeling the PLL
, the feedback nature of the PLL causes the VCO to synchronize
V
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(t)
i
e
Figure 1. Block Diagram of Phase-Locked Loop
COMPARATOR
CONTROLLED
I
OSCILLATOR
o
and thus keep the PLL in lock. This selfcorrecting
VOLTAGE
e
PHASE
(t) that is related to the phase and the frequency
K
K
d
o
V
i
o
o
(t)
1
V
e
V
(t)
e
d
(t)
e
where
LOW–PASS
d
O
I
(t) forces the VCO frequency to
FILTER
’) which is known as the
F
(s)
O
shown in Figure 1. When
I
V
1
is sufficiently close to
d
(t)
f
A
d
to shift the
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OUTPUT
SIGNAL
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O
’ (or
(1)
O
2
the difference frequency component ((
output of the phase comparator contains only a DC component. The
low-pass filter removes the sum frequency component (
passes the DC component which is then amplified and fed back to
the VCO. Notice that when the loop is in lock, the difference
frequency component is always DC, so the lock range is
independent of the band edge of the low-pass filter.
LOCK AND CAPTURE
Consider now the case where the loop is not yet in lock. The phase
comparator again mixes the input and VCO signals to produce sum
and difference frequency components. However, the difference
component may fall outside the band edge of the low-pass filter and
be removed along with the sum frequency component. If this is the
case, no information is transmitted around the loop and the VCO
remains at its initial free-running frequency. As the input frequency
approaches that of the VCO, the frequency of the difference
component decreases and approaches the band edge of the
low-pass filter. Now some of the difference component is passed,
which tends to drive the VCO towards the frequency of the input
signal. This, in turn, decreases the frequency of the difference
component and allows more information to be transmitted through
the low-pass filter to the VCO. This is essentially a positive
feedback mechanism which causes the VCO to snap into lock with
the input signal. With this mechanism in mind, the term “capture
range” can again be defined as ‘the frequency range centered about
the VCO initial free-running frequency over which the loop can
acquire lock with the input signal’. The capture range is a measure
of how close the input signal must be in frequency to that of the
VCO to acquire lock. The “capture range” can assume any value
within the lock range and depends primarily upon the band edge of
the low-pass filter together with the closed-loop gain of the system.
It is this signal capturing phenomenon which gives the loop its
frequency-selective properties.
It is important to distinguish the “capture range” from the “lock
range” which can, again, be defined as ‘the frequency range usually
centered about the VCO initial free–running frequency over which
the loop can track the input signal once lock has been achieved’.
When the loop is in lock, the difference frequency component at the
output of the phase comparator (error voltage) is DC and will always
be passed by the low-pass filter. Thus, the lock range is limited by
the range of error voltage that can be generated and the
corresponding VCO frequency deviation produced. The lock range
is essentially a DC parameter and is not affected by the band edge
of the low-pass filter.
THE CAPTURE TRANSIENT
The capture process is highly complex and does not lend itself to
simple mathematical analysis. However, a qualitative description of
the capture mechanism may be given as follows. Since frequency is
the time derivative of phase, the frequency and the phase errors in
the loop can be related as
where
signal and VCO frequencies and
the input signal and VCO signals.
If the feedback loop of the PLL were opened between the low-pass
filter and the VCO control input, then for a given condition of
fixed frequency
I
the phase comparator output would be a sinusoidal beat note at a
is the instantaneous frequency separation between the
d
d
t
e
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. If
I
and
O
were sufficiently close in
e
is the phase difference between
I
x
O
) is zero; hence, the
Application note
AN178
I
+
O
O
) but
(2)
and