Page 54 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL ANALYSIS Ronald E. Best 40
for phase detectors with voltage output or

for phase detectors with current output,
hence the mathematical model of the phase detector is simply a zero-order block with
gain K or K (also referred to as “gain block”). The transfer function of the phase
P
d
detector is therefore given by

(3.2)

(3.3)

where s is the Laplace operator. (App. B provides an introduction to Laplace transforms.)
The transfer functions of the loop filter have already been derived in Sec. 2.5 and are
denoted here with F(s). Let’s now look at the transfer function of the VCO. As stated in
Eq. (1.1), the angular frequency of the VCO is given by

(3.4)

−1 −1
where K is called VCO gain (dimension: rad s V ). The model of the VCO should
0
yield the output phase θ , however, and not the output frequency ω . By definition, the
2
2
phase θ is given by the integral over the frequency variation Δω
2 2

(3.5a)

In the Laplace transform, integration over time corresponds to division by s, so the
Laplace transform of the output phase θ is given by
2

(3.5b)

The transfer function of the VCO is therefore given by

(3.6)

For phase signals, the VCO simply represents an integrator.
Lastly, we must also know the mathematical model of the (optional) divide-by-N
counter (cf. Fig. 2.1). Because this counter divides the frequency created by the VCO by a