Global Positioning System Reference
In-Depth Information
The second type is a frequency-modulated signal
ω
s
2
θ
i
(t)
=
ωt
or
θ
i
(s)
=
(
8
.
13
)
These two types of signals will be discussed in the next two sections.
8.3 FIRST-ORDER PHASE-LOCKED LOOP
(
1-4
)
In this section, the first-order phase-locked loop will be discussed. A first-order
phase-locked loop implies the denominator of the transfer function
H
(
s
)isafirst-
order function of
s
. The order of the phase-locked loop depends on the order of
the filter in the loop. For this kind of phase-locked loop, the filter function is
F(s)
=
1
(
8
.
14
)
This is the simplest phase-locked loop. For a unit step function input, the corre-
sponding transfer function from Equation (8.9) becomes
k
0
k
1
H(s)
=
(
8
.
15
)
s
+
k
0
k
1
The denominator of
H
(
s
) is a first order of
s
.
The noise bandwidth can be found as
∞
∞
(k
0
k
1
)
2
df
ω
2
(k
0
k
1
)
2
2
π
dω
B
n
=
=
+
(k
0
k
1
)
2
ω
2
+
(k
0
k
1
)
2
0
0
tan
−
1
ω
k
0
k
1
∞
0
=
(k
0
k
1
)
2
2
πk
0
k
1
k
0
k
1
4
=
(8.16)
With the input signal
θ
i
(s)
=
1
/s
, the error function can be found from
Equation (8.10) as
1
s
+
k
0
k
1
(s)
=
θ
i
(s)H
e
(s)
=
(
8
.
17
)
The steady-state error can be found from the final value theorem of the Laplace
transform, which can be stated as
lim
y(t)
=
lim
s
0
sY(s)
(
8
.
18
)
t
→∞
→
Using this relation, the final value of
(
t
) can be found as
s
lim
(t)
=
lim
s
0
s(s)
=
lim
s
k
0
k
1
=
0
(
8
.
19
)
s
+
→∞
t
→
→
0
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