Page 249 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 148
By applying the addition theorem of trigonometric functions once again, we obtain

(6.27)

Clearly the capacitor voltage consists of three terms; the first one is the “carrier” at
radian frequency ω , and the other terms represent upper and lower sidebands at radian
0
frequencies ω + ω and ω − ω , respectively. These sidebands are created by the noise
0 m 0 m
signal u added to the comparator threshold U . The side-bands create a phase perturbation of
n
c
the signal across the capacitor. Applying the same theory we used to compute the phase noise
of the reference oscillator [cf. Sec. 6.7.1 and Eq. (6.7)], the phase perturbation can be
computed from

The amplitude of one sideband is , where U is the
n,rms

rms value of the noise signal. The rms value of one sideband is therefore and

its power is the rms value squared—in other words, . Because we have two

sidebands, the total noise power is twice as much: . Because the carrier power

is the phase perturbation becomes

(6.28)

This is the phase perturbation of the capacitor voltage U cap 1
, caused by a noise signal
having frequency f . Noise signals are normally broadband signals, thus we define by
m
the power spectral density of the noise signal measured in W/Hz (we assume, as

usual, that the signal is applied to a load resistor of 1 Ω). Then · B, B = 1 Hz is
the noise power in a band that is 1 Hz wide at frequency f . We now are in the position to
m