Global Positioning System Reference
In-Depth Information
The equation for a third-order PLL is similar [10]:
θ
ωτ
∆
()
στ
=
225
.
(
dimensionless units of
∆
ff
)
(5.11)
A
L
2
(), has already been determined for the oscillator for
If the Allan variance,
στ
A
the short-term gate time,
τ
, then the Allan deviation-induced jitter in degrees,
θ
A
=
2
360
()changes
very little for the short-term gate times involved. These gate times must include the
reciprocal of the range of noise bandwidths used in the carrier loop filters,
∆θ
/2
π
, can be computed from the previous equations. Usually
στ
A
1/
B
n
.A
short-term gate-time range of 5 ms to 1,000 ms should suffice for all PLL applica-
tions. Rearranging (5.10) using these assumptions, the equation for the sec-
ond-order loop is:
τ =
()
στ
f
A
L
θ
=
144
(degrees)
(5.12)
A
2
B
n
Rearranging (5.11) using these assumptions, the equation for the third-order
loop is:
()
στ
f
A
L
θ
=
160
(degrees)
(5.13)
A
3
B
n
For example, assume that the loop filter is third-order with a noise bandwidth,
B
n
=
18 Hz, tracking the L1 signal, and the Allan deviation is specified to be
σ
A
(
τ
)
=
1
×
10
−10
or better for gate times that include
τ =
1/
B
n
=
56 ms. The phase jitter contri-
bution due to this error is
1.40° or less. Obviously, a reference oscillator with a
short-term Allan deviation characteristic that is more than an order of magnitude
worse than this example will cause PLL tracking problems.
Figure 5.23 graphically portrays the sensitivity of a third-order PLL to changes
in short-term Allan deviation performance of the reference oscillator, especially as
the noise bandwidth,
B
n
, is narrowed. The objective of narrowing the bandwidth is
to reduce the thermal noise error to improve the tracking threshold. However, as
Figure 5.23 illustrates, the Allan deviation effects begin to dominate at the narrower
noise bandwidths. This effect is usually the primary source of aided GPS receiver
narrowband PLL tracking problems, assuming that the external velocity aiding
accuracy is not the limiting factor. However, even for an unaided GPS receiver, a ref-
erence oscillator with a poor Allan deviation characteristic, say a
θ
A
3
=
×
10
−9
, will prevent reliable PLL operation. Therefore, the oscillator specification for
Allan deviation is important for all GPS receiver designs.
∆
f
/
f
of less than 1
5.6.1.4 Dynamic Stress Error
The dynamic stress error is obtained from the steady state error formulas shown in
Table 5.6. This error depends on the loop bandwidth and order. The maximum
dynamic stress error may be slightly larger than the steady state error if the loop fil-
ter response to a step function has overshot, but the steady state error formula will
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