Page 243 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 145
We note that the phase frequency response H(f) acts as a lowpass filter onto the phase jitter
caused by the reference oscillator. To minimize VCO output jitter, the loop bandwidth (B )
L
must be made as small as possible. But this bandwidth cannot be chosen arbitrarily low,
because a low B also means a low natural frequency ω —in other words, a large lock-in
L
n
time.

Phase noise created by the VCO
In Sec. 6.7.1, we investigated the phase jitter introduced by the reference oscillator. Checking
the phase jitter model of Fig. 6.14 once again, we recognize that the VCO is nothing else than
an oscillator, too, hence it will introduce additional phase jitter into the loop. This phase noise
(denoted θ n,VCO in the block diagram) enters the loop at the input of the summator labeled Σ.
If there were no feedback, phase noise θ would simply add to the output phase noise
n,VCO
θ n,out . Because we have a feedback path, we must compute the closed-loop gain G from the
n
insertion point of the VCO phase noise θ to the output terminal θ . For G , we obtain
n,VCO n,out n
the very simple result

(6.18)

in other words, it is identical with the error frequency response as defined in Eq. (3.20). As
can be seen from the Bode diagram for H (ω) in Fig. 3.3, H (ω) is a highpass function. The
e e
PSD of phase noise contributed by the VCO is given by

(6.19)

(f ) = PSD of output
(f ) = PSD of phase jitter introduced by the VCO, S
with S θθ,VCO m θθ,out m
phase jitter due to VCO phase jitter, and f = frequency offset from center frequency f . The
m 0
bandwidth of highpass function H (f) is the same as the bandwidth of the lowpass function H
e
(f), because H (f) = 1 − H(f) [c.f. Eq. (3.9)]. If we assume that most of the VCO phase noise is
e
flicker noise, most of its power is concentrated around the center frequency f . To minimize
0
output phase jitter, it would be optimum to make the PLL bandwidth as large as possible. This
is in clear contradiction to the conclusion in Sec. 6.7.1 where we stated that the loop
bandwidth should be made small in order to reduce reference phase jitter.
As we have seen in Sec. 2.6, there are two entirely different types of VCOs in use:
relaxation oscillators and resonant oscillators. For the latter, the PSD of oscillator phase noise
has been analyzed in Sec. 6.7.1, hence we can adopt these results to compute the overall
output phase noise S θθ,out m
(f ) immediately.
For relaxation oscillators, however, no phase noise model is known at present. The author
tried therefore to develop such a model. This will be presented next.
Noise model for relaxation type VCOs. The operating principle of the relaxation oscillator
has been explained in Sec. 2.6.1 (refer to Fig. 2.23). It was shown that the voltage U across
cap
capacitor C has a triangular waveform, as shown by the solid curve in Fig. 6.19. The