Page 262 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 155
where P is the power of the synthesized signal and P sb1 is the power of the first spurious
s
sideband. Mostly, a logarithmic quantity S is specified:
1(dB)

(6.35)

If the EXOR or the JK-flipflop is used, an approximation for S is given by 15
1(dB)

(6.36)

Here f denotes ripple frequency. F(2πf ) is the gain of the loop filter at the ripple frequency,
r r
and U is the supply voltage of the phase detector. In the case of the EXOR gate, f equals
r
B
twice the reference frequency.
When the JK-flipflop is used, however, f is identical with the reference frequency.
r
For the PFD with voltage output, another approximation has been found: 15

(6.37)

Here, the ripple frequency f is usually a subharmonic of the reference frequency f , as
r
ref
described earlier. τ is the duration of PFD output pulses, as shown in Fig. 6.21. For τ, we have
approximately

(6.38)

with τ = R C [cf. Fig. 6.21].
1 p
2
The charge pump PFD is optimum with respect to spurs. If the current sources were
perfectly matched, there would be theoretically no spurs at all. An imbalance of the current
sources leads to a spur, however, at the reference frequency. The following analysis leads to
an approximate formula for the power of this spur.
Assume that the imbalance of the current sources is δ—that is, one of the current sources of
the PFD (cf. Fig. 2.16) supplies current I , the other (1−δ)I . When the overlapping interval
P
P
(as defined in Fig. 6.22) is T , a net charge of Q = δ I T ov flows into the load in every
P
ov
reference cycle. This charge must be compensated for by a finite phase error. When the sink
current is the larger one, Q becomes negative, and the phase error must become positive so a
positive charge will be supplied in the interval T , as shown in Fig. 6.23. Because the area
comp
under the positive pulse must equal the area under the negative pulse, we have

(6.39)