Global Positioning System Reference
In-Depth Information
where
t
θ
f
(t)
=
k
1
u(t)V
o
dt
(
8
.
3
)
0
The Laplace transform of
θ
f
(t)
is
V
o
(s)
k
1
s
θ
f
(s)
=
(
8
.
4
)
From Figure 8.1b the following equations can be written.
V
c
(s)
=
k
0
(s)
=
k
0
[
θ
i
(s)
−
θ
f
(s)
]
(8.5)
V
o
(s)
=
V
c
(s)F (s)
(8.6)
θ
f
(s)
=
V
o
(s)
k
1
s
(8.7)
From these three equations one can obtain
V
c
(s)
k
0
V
o
(s)
k
0
F(s)
=
sθ
f
(s)
k
0
k
1
F(s)
or
(s)
=
θ
i
(s)
−
θ
f
(s)
=
=
θ
f
(s)
1
s
k
0
k
1
F(s)
θ
i
(s)
=
+
(8.8)
where
(s)
is the error function. The transfer function
H
(
s
) of the loop is
defined as
θ
f
(s)
θ
i
(s)
=
k
0
k
1
F(s)
H(s)
≡
(
8
.
9
)
s
+
k
0
k
1
F(s)
The error transfer function is defined as
(s)
θ
i
(s)
=
θ
i
(s)
θ
f
(s)
θ
i
(s)
−
s
s
+
k
0
k
1
F(s)
H
e
(s)
=
=
1
−
H(s)
=
(
8
.
10
)
The equivalent noise bandwidth is defined as
(
1
)
∞
2
df
B
n
=
|
H(jω)
|
(
8
.
11
)
0
where
ω
is the angular frequency and it relates to the frequency
f
by
ω
2
πf
.
In order to study the properties of the phase-locked loops, two types of input
signals are usually studied. The first type is a unit step function as
=
1
s
θ
i
(t)
=
u(t)
or
θ
i
(s)
=
(
8
.
12
)
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