TABLE OF CONTENTS If we assume that the PLL has locked and stays locked for the near future, we can develop a linear mathematical model for the system. As will be shown in this section, the mathematical model is used to calculate a phase-transfer function H( s) that relates the phase ? 1 of the input signal to the phase ? 2 ? of the output signal (of the down-scaler):
| (2.30) |  |
where 
are the Laplace transforms of the phase signals ? 1( t) and ? 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 
is used for the Laplace transform of phase ? 2 ?.) H( s) is called phase-transfer function. To get an expression for H( s) we must 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. 2.4.1.
As derived in Sec. 2.3.1, in the locked state the output signal u d of the phase detector can be approximated by
hence the mathematical model of the phase detector is simply a zero-order block with gain K d (also referred to as a gain block). The transfer function...
