14
Application Information
A block diagram of the basic phase locked loop is shown in
Figure 21.
Loop Gain Equations
A linear control system model of the phase feedback fo r a
PLL in th e locked st ate is shown in Fi gure 22 . The open loop
gain i s the produc t of the phase comparator gain (Kφ), t he
VCO gain (KVCO/s), and the loop f il ter gain Z(s) divided by
the gain of the feed back counter modul us (N). The passive
loop filt er configuration used is displayed in Figure 23, while
the complex impedance of the filter is given in Equation 2.
The time constants which det ermine the pole and zero
frequencies of the filter transfer function can be defi ned as:
and
The 3rd order PLL Open Loop Gain can be calcul ated in
terms of frequency, ω, t he filter time const ants T1 and T2,
and the design constants Kφ, KVCO, and N.
From Equation 3 we can see that the phase term will be
dependent on the single pole and zero such that the phase
margin is determined in Equati on 5.
A plot of the magnitude and phase of G(s) H(s) for a stable
loop, is sh own in Figure 24 with a solid t race. The parameter
φP shows the amount of ph ase margin t hat ex ists at th e point
the gain drops below zero (the cutoff f requency wp of the
loop). In a critically damped system, the amount of phase
margin would be approximately 45 degrees.
If we were now to redefine the cut off frequency, wp’, as
double the frequency which gave us our original loop
bandwidth, wp, the loop response time would be
approximately halved. Because the filter attenuation at the
comparison frequency also diminishes, the spurs would have
increased by approximately 6dB. In the proposed Fastlock
sch e me, th e h igh er spu r lev el s an d w ide r loo p f il te r c ond i ti on s
would exist only during the initial lock-on phase - just long
enough to reap the benefits of locking faster. The objective
would be to open up the loop bandwidth but not introduce any
additional complications or compromises related to our
original design criteria. We would ideally like to momentarily
shift the curve of Figure 24 over to a different cutoff frequency,
illustrated by the dotted line, without affecting the relative
open loop gain and phase relationships. To maintain the
same gain/phase relationship at twice the original cutoff
frequency, other terms in the gain and phase Equations 4 and
5 will have to compensate by the corresponding “1/w” or 1/w2”
factor. Examination of Equations 3 and 5 indicates the
damping resistor variable R2 could be chosen to compensate
the “w” terms for the phase margin. This implies that another
resistor of equal value to R2 will need to be switched in
fOUT
VCO
LOOP
Z(s)
CHARGE
PUMP
FILTER
DO
fP
fR
PHASE
DETECTOR
φR
φP
Φ
1/N
MAIN
DIVIDER
REFERENCE
FREQUENCY
fREF
FREQUENCY
SYNTHESIZER
1/R
REFERENCE
DIVIDER
CRYSTAL
REFERENCE
VP
fIN
FIGURE 21 . BASIC CHARGE PUMP PHASE LOCKED LOOP
KVCO
s
Z(s)
Kφ
1/N
∑
ΘE
ΘR
ΘI
-
+ΘO
FIGURE 22. PLL LINEAR MODEL
C2
R2
C1
VCODO
FIGURE 23. PASSIVE LOOP FILTER
(EQ. 1)
Open loop gain H(s) G(s) ΘIΘE
⁄==
Kφ Z(s) K VCO Ns⁄=
Zs() sC2 R2•()1+
s2C1 C2 R2••()sC1 sC2++
-----------------------------------------------------------------------------------= (EQ. 2)
T1 R2 C1 C2•
C1 C2+
----------------------•=(EQ. 3A)
T2 R2 C2•=(EQ. 3B)
Gs() Hs()•s = j w •
KφKVCO 1jωT2•+()•–
ω2C1 N 1 jωT1•+()•
-------------------------------------------------------------------T1
T2
-------•=(EQ. 4)
φω() tan 1–ωT2•()tan 1–ωT1•()–180o
+= (EQ. 5)
HFA3524