AN178 Philips, AN178 Datasheet - Page 9

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

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DataSheet
Philips Semiconductors
Figure 9 is a plot of the VCO phase response and the phase error
transient for various damping factors. Note from this figure that an
underdamped system has overshoot which can cause the loop to
break lock if this overshoot is too large. The critical condition for
maintaining lock is to keep the phase error within the dynamic range
for the phase comparator of - /2 to 2 radians. For the
underdamped case, the peak phase-error overshoot is
which must be less than /2 to maintain lock. Lock can also be
broken for the overdamped and critically-damped loops if the input
phase shift is too large where the phase error exceeds
The analysis and equations given are based upon the small-signal
model of Figure 4. If the signal amplitudes become too large, one or
more functional blocks in the system can saturate, causing a slew
rate type limiting action that may break lock.
The transient change in the VCO frequency due to the unit
step-of-phase input can be found by taking the time derivative of
Equation 41 or alternatively by finding the inverse Laplace transform
of
which is
Unit Step-of-Frequency Input
This type of input occurs when the input frequency is
instantaneously changed from one frequency to another as is done
1988 Dec
4
U
Modeling the PLL
.com
Figure 10. Input Signal for a Unit Step-of-Frequency at
e
Figure 9. VCO Phase and Loop Phase Error Transient
o
O
(max)
(s)
(t)
Responses for Various Damping Factors
s
n
o
e
e
1
(s)
2
s
n
Constant Phase
t
2
sin
1
2
n
t 1
n
n
2
s
2
n
2
2
/2 radians.
SL01020
SL01019
DataSheet4U.com
(46)
(47)
(48)
9
in FSK and modem applications. For this input, as shown in
Figure 10,
The VCO output phase is
The time expression for the VCO phase change is
for
The time expression for the VCO frequency change for a unit
step-of-frequency input is the same as the time response VCO
phase change due to a step-of-phase input (Equation 41), or
for
Unit Ramp-of-Frequency Input
This form of input signal represents sweeping the input frequency at
a constant rate and direction as shown in Figure 11. The amplitude
and phase of the input remain constant; the input frequency
changes linearly with time. Since the input signal to the PLL model
is a phase, a unit ramp-of-frequency appears as a phase
acceleration type input that can be mathematically described as
The VCO output phase change is
The time expression for the VCO phase change is
where
and y is given in Equation 42.
O
(t) for frequency step input =
Figure 11. Input Signal for a Unit Ramp-of-Frequency Input
o
(t)
i
o
i
o
o
(s)
o
(s)
(s)
1.
1.
(s)
(s)
(t)
t
arc tan
s
s
1
1
1
s
s
t
2
2
3
2
2
3
2
(s
(s
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n
2
2
x e
e
1
2 t
1
(1
n
2
2
e
1
n
4
n t
n
n
2
2
n
n
2
n
1
2
1
t
sin(
2
s
s
2
sin (
n
2
n
2
n
2
2
n
[2 (1
o
t
n
(t) for phase step input. Thus
2
t
2
n
n
n
2
sin (
t 1
2
4
)
1
2
n
4
n
2
t 1
n
2
2
1
2
)
Application note
2
AN178
2
SL01021
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(55)