Tracking GPS Signals - Fundamentals of Global Positioning System Receivers: A Software Approach - page 164

With the input signal θ i (s) = ω/s 2 , the error function is

ω
s

1
s + k 0 k 1

(s) = θ i (s)H e (s) = ( 8 . 20 )

The steady-state error is

ω ω

k 0 k 1

lim
t →∞

(t) = lim
s 0 s(s) = lim

k 0 k 1 =

( 8 . 21 )

s +

→ →

s 0

This steady-state phase error is not equal to zero. A large value of k 0 k 1 can make
the error small. However, from Equation (8.15) the 3 dB bandwidth occurs at
s

k 0 k 1 . Thus, a small final value of ( t ) also means large bandwidth, which
contains more noise.

=

8.4 SECOND-ORDER PHASE-LOCKED LOOP ( 1-4 )

A second-order phase-locked loop means the denominator of the transfer function
H ( s ) is a second-order function of s . One of the filters to make such a second-
order phase-locked loop is

sτ 2 + 1

F(s) = ( 8 . 22 )

sτ 1

Substituting this relation into Equation (8.9), the transfer function becomes

k 0 k 1 τ 2 s
τ 1

k 0 k 1
τ 1

+

ω n

2 ζω n s +

= ≡

H(s) ( 8 . 23 )

k 0 k 1 τ 2 s
τ 1

k 0 k 1
τ 1

s 2 + 2 ζω n s + ω n

s 2 + +

where ω n is the natural frequency, which can be expressed as

k 0 k 1
τ 1

ω n = ( 8 . 24 )

and ζ is the damping factor, which can be shown as

k 0 k 1 τ 2
τ 1

ω n τ 2
2

2 ζω n = or ζ = ( 8 . 25 )

The denominator of H ( s ) is a second order of s .