U. Glaeser

db(T( )/G O)
π ω
ph(T( )/ )
ω
FIGURE 10.5 PLL loop gain magnitude and phase (without C 2).
The closed-loop response can be derived from the open-loop response by considering the feedback
signal. In the closed-loop system, the output phase, P O(s), is related to the input phase, P I(s), by
© 2002 by CRC Press LLC
P O(s) = (P I(s) − P O(s)/N) ⋅ H(s)
where N is the feedback clock divider value. The closed-loop response is then given by
or, equivalently, by
200
150
100
50
0
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-50
-100 | | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | ||||||
10-4 10-3 10-2 10-1 100 101 102 103 104 0.0
-0.2
-0.4
-0.6
-0.8
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where ζ, defined as the damping factor, is given by
•R•C
•R•C
-1.0 | | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | |||||| | | | ||||||
10-4 10-3 10-2 10-1 100 101 102 103 104 |
PO()�P s I s
PO ()�P s I s
ζ = 1�2 ⋅ (1/ N ⋅ I CH ⋅ K V ⋅ R 2 ⋅ C) 0.5
and ω N, defined as the loop bandwidth (rad/s), is given by
ω
ω N = 2 ⋅ ζ �(R ⋅ C)
The loop bandwidth and damping factor completely characterize the closed-loop response. The PLL
is critically damped with a damping factor of one and overdamped with damping factors greater than one.
Treating the PLL as a standard second order system makes it much easier to analyze. The time domain
impulse, step, and ramp responses are easily derived from the frequency domain closed-loop response.
Equations for these responses are summarized in Table 10.2. The peak values of these responses are very
useful in estimating the amount of frequency overshoot and the amount of supply and substrate noise
ω
() = 1�( 1/N + s�H() s)
= N⋅1+ s⋅ C⋅ R/
1 s C R s 2 ( + ⋅ ⋅ + / ( ICH/C ⋅ KV/N) )
() = N⋅( 1 + 2 ⋅ζ⋅( s/ω N)�
( 1 2 ζ ( s/ω N)
( s/ω N)
2 +
⋅ ⋅ + )