AN178 Philips, AN178 Datasheet - Page 4

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

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Philips Semiconductors
Unlocked State (
When the two frequencies to the phase comparator are not
synchronized, the loop is not locked. Furthermore, the phase angle
difference
it can be eliminated by appropriately choosing the time origin.
Using trigonometric identities, Equation 5 can be rewritten as
When v
frequency component is removed, leaving
where K
the VCO appears as
This equation shows that a beat frequency effect is established
between
from
through the filter. If the amplitude of V
signal limiting or saturation does not occur, the VCO output
frequency will be shifted from
established where
If lock cannot be established, then either V
VCO to produce the necessary
dynamic range of the VCO, i.e.,
no lock conditions are:
1. Increase Vl either internally or externally to the loop by providing
2. Increase the internal loop gain by adjusting upward (larger -3dB
3. Shift
Locked State (
When
the phase comparator for
The low-pass filter removes the high frequency, AC component of
v
After amplification the DC voltage driving the VCO and maintaining
lock within the loop is
Suppose
frequency
must be 90 . Thus V
phase error between
angle of 90 . If
a change in
non-zero value to correct
maintained with
1988 Dec
4
e
(t), leaving only the DC component. Thus,
U
Modeling the PLL
additional amplification.
frequency) the response of the low-pass filter.
leads to the second case where
v
v
v
v
v
v
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e
f
d
e
f
d
(t)
I
(t)
O
(t)
(t)
(t)
(t)
e
’ in proportion to the signal amplitude (AK
I
2
(t) is passed through the low-pass filter, F(s), the sum
and
is a constant. After amplification, the control voltage for
O
I
I
’ closer to the expected
e
and
O
and
K
O
K
in Equations 4 and 5 is meaningless lor this case since
K
AK
K
K
V
’. For this case, V
e
2
2
1
1
1
D
O
V
V
from 90 . V
V
V
V
2
2
2
I
I
are frequency synchronized, the output signal from
I
I
V
V
I
I
V
O
I
V
=
V
V
O
I
=
I
O
I
, causing the VCO’s frequency to deviate by
O
V
O
O
O
changes slightly from
cos(
are perfectly synchronized to the free-running
O
cos
AK
(sin t) sin ( t
O
[cos
cos (
O
[cos(
D
i
O
)
cos (
and
O
2
. The phase error will be shifted by some
is proportional to the phase difference or
V
)
I
e
I
e
V
O
I
D
I
=
O
; under this condition frequency lock is
I
o
will adjust and settle out to some
cos
I
centered about a reference phase
O
cos (2 t
O
D
O
’ by some
=
I
)t]
will be zero, indicating that
><
O
O
e
)t
deviation or
)t
O
I
I
. Establishing frequency lock
and a phase shift of
l
)t
=
is sufficiently large and if
O
’
e
O
)
l
O
.
is too small to drive the
’, the first effect will be
e
. Remedies for these
)]
until lock is
2
I
V
is beyond the
I
V
O
) passing
e
DataSheet4U.com
is
e
(6)
(7)
(8)
(9)
(10)
(11)
(12)
4
amount
can be simplified by redefining
where
departure from this reference value. Now the VCO control voltage
becomes
Since the sine function is odd, a momentary change in
information about which way to adjust the VCO frequency to correct
and maintain the locked condition. The maximum range over which
In addition to being an error signal, V
output of an FM input applied as v
characteristic. Thus, FM demodulation can be accomplished with
the PLL without the inductively-tuned circuits that are employed with
conventional detectors.
DETERMINING PLL MODEL PARAMETERS
Since the PLL is basically an electronic servo loop, many of the
analytical techniques developed for control systems are applicable
to phase-locked systems. Whenever phase lock is established
between v
predict the performance of the PLL system. Here
represent the phase angles associated with the input/output
waveshapes, respectively; F(s) represents a generalized voltage
transfer function for the low-pass filter in the s complex frequency
domain; and K
comparator and VCO, respectively, each having units as shown.
The 1/s term associated with the VCO accounts for the inherent 90
phase shift in the loop since the VCO converts a voltage to a
frequency and since phase is the integral of frequency. Thus the
VCO functions as an integrator in the feedback loop.
Specific values of K
general purpose PLLs can be found in the sections describing the
particular loop of interest. However, sometimes it may be desired to
determine these conversion gains exactly for a specific device. The
measurement scheme shown in Figure 5 can be used to determine
K
filters is to extract the fundamental sinusoidal frequency component
of their square wave inputs for application to the Gain-Phase Meter.
If the input signal from the Function Generator is sinusoidal, then the
first Khron-Hite filter may be eliminated. It is recommended to use
high impedance oscilloscope probes so as to not distort the input of
VCO waveshapes, thereby potentially altering their phase
relationships. The frequency counter can be driven from the scope
as shown, or connected directly to the input or VCO, provided its
input impedance is large.
e
f
d
S)
range from 0 to 180 .
changes can be tracked is -90 to +90 . This corresponds to a
and Ko for a loop under lock. The function of the Khron-Hite
v
e
D
r
, is the inherent, reference phase shift of 90 and
i
(t) and v
s)
from the reference phase angle of 90 . This concept
AK
e
(S)
r
d
Figure 4. Linear Model of PLL System
1
V
and K
AK
I
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V
K
o
O
d
2
d
(t) the linear model of Figure 4 can be used to
v
cos(
RADIAN
and K
I
VOLTS
o
V
K
S
o
are conversion gains of the phase
O
sin
r
o
VOLT SEC.
RADIANS
for all of Philips Semiconductors
V
E
(S)
e
)
as
in
(t) assuming a linear VCO
F
D
s
VOLT
VOLT
represents the demodulated
V
F
(S)
i
and
Application note
A
AN178
VOLTS
VOLT
SL01014
o
contains
V
is the
D
(S)
(13)
(14)