title: Frequency Synthesis by Phase Lock
author:  William F. Egan 
description: A through discussion of frequency synthesizers, including designand effects on systems Working engineers who design, specify, use,or test frequency synthesizers need to develop an intimateunderstanding of how these devices operate and their effects on thesystems in which they are embedded. Frequency Synthesis by PhaseLock, Second Edition, offers complete coverage that includes bothnormal control system design and effects that occur due to samplingwhen bandwidths are pushed. While the discussion emphasizesphase-locked synthesizers, direct and digital synthesizers arecovered as well. In addition to the usual discussion ofsecond-order loops, this book describes characteristics of animportant class of third-order loops and state-space analysis ofloops of arbitrary order. It uses Matlab scripts (available fordownloading via ftp) to produce computer-aided analyses, includingcomplex nonlinear simulations of loops in the acquisition process;and it includes a significant volume of material on phase noise,its effects in synthesizer loops, and its impact on systemsemploying synthesizers. An important reference of rare power andclarity, Frequency Synthesis by Phase Lock, Second Edition,features:<br />* Three new chapters covering architectures, sampling effects, andcomputer-aided engineering (CAE)<br />* Multicurve graphs of transient and modulation responses forsecond-and third-order loops<br />* Graphs of phase noise from 28 oscillators and 19 frequencydividers; noise theory and curves for IC oscillators<br />* Charge-pump phase-frequency detectors extensively covered<br />* Fractional-N, including DSP for improved spectrums<br />* Multiple loops, including offset references
zlib_text: Chapter 5
  Phase System Parallel Coupling
  5.1.
  Models and Modes
  The structural scheme of an ensemble of two phase-locked loops con-
  nected in parallel (PPLL) is shown in Fig. 5.1. Here, the reference
  signal for both voltage controlled oscillators of the ensemble is signal
  S0(t). It comes to the phase discriminators PD1 and PD2, where it
  is compared with signals S1(t) and S2(t) of the voltage controlled
  oscillators G1 and G2, respectively. At the outlet of PD1 and PD2
  voltages u1 and u2 are produced, which are proportional to the dif-
  ference of oscillation phases of the voltage controlled oscillator and
  the reference signal. Further, the phase detuning signals u1 and u2
  come to the summators Σ, where it is summarized with the signals
  of the couplings u′
  1 = κu2 and u′
  2 = δu1. Signals ˜u1 = u1 + κu2 and
  ˜u2 = u2 + δu1 are used for managing the frequencies of generators
  G1 and G2 with control circuits CC1 and CC2. Operation equations
  describing such system dynamics are as follows [4]:
  pϕ