Page 53 - Phase-Locked Loops Design, Simulation, and Applications

Source : Phase-Locked Loops: Design, Simulation, and Applications, Sixth
Edition Ronald E. Best 39

Mixed-Signal PLL Analysis

PLL Performance in the Locked State

Having discussed the components of a mixed signal PLL system, we will now analyze the
dynamic performance of this type of PLL. The symbols used in the following have been
defined in Fig. 2.1.
If we assume that the PLL has locked and will stay locked in the near future, we can
develop a linear mathematical model for the system. As will be shown in Sec. 3.3, the
mathematical model is used to calculate a phase-transfer function H(s), which relates the
phase θ of the input signal to the phase θ ′ of the output signal (of the down scaler):
1 2

(3.1)

where Θ (s) and Θ ′(s) are the Laplace transforms of the phase signals θ (t) and θ ′
1
2
1
2
(t), respectively. (Note that we are using lowercase symbols for time functions and
uppercase symbols for their Laplace transforms throughout the text; this also applies to
Greek letters. Furthermore, the symbol Θ ′(s) is used for the Laplace transform of phase
2
θ ′.) H(s) is called phase-transfer function. To get an expression for H(s), we must
2
know the transfer functions of the individual building blocks in Fig. 2.1. This transfer
function will be calculated from a mathematical model that will be derived in Sec. 3.2.

The Mathematical Model for the Locked State

As derived in Sec. 2.4, in the locked state the output signal u of the phase detector can be
d
approximated by

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