scholarly journals On the Gardner Problem for the Phase-Locked Loops

2019 ◽  
Vol 489 (6) ◽  
pp. 541-544
Author(s):  
N. V. Kuznetsov ◽  
M. Y. Lobachev ◽  
M. V. Yuldashev ◽  
R. V. Yuldashev
Keyword(s):  
Phase Space ◽  
Order System ◽  
Phase Locked Loops ◽  
Stability Criteria ◽  
Third Order ◽  
Classical Stability ◽  
Cylindrical Phase ◽  
Lock In ◽  
Cylindrical Phase Space

This report shows the possibilities of solving the Gardner problem of determining the lock-in range for multidimensional phase-locked loops systems. The development of analogs of classical stability criteria for the cylindrical phase space made it possible to obtain analytical estimates of the lock-in range for third-order system.

FRACTALS AND CLOSURES OF LINEAR DYNAMICAL SYSTEMS EXCITED STOCHASTICALLY BY TEMPORAL INPUTS

2001 ◽  
Vol 11 (03) ◽  
pp. 755-779 ◽  
Author(s):  
RYOICHI WADA ◽  
KAZUTOSHI GOHARA
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Invariant Set ◽  
Poincaré Section ◽  
Poincare Section ◽  
Linear Dynamical Systems ◽  
Cylindrical Phase ◽  
Iterated Functions ◽  
Cylindrical Phase Space

Fractals and closures of two-dimensional linear dynamical systems excited by temporal inputs are investigated. The continuous dynamics defined by the set of vector fields in the cylindrical phase space is reduced to the discrete dynamics defined by the set of iterated functions on the Poincaré section. When all iterated functions are contractions, it has already been shown theoretically that a trajectory in the cylindrical phase space converges into an attractive invariant set with a fractal-like structure. Calculating analytically the Lipschitz constants of iterated functions, we show that, under some conditions, noncontractions often appear. However, we numerically show that, even for noncontractions, an attractive invariant set with a fractal-like structure exists. By introducing the interpolating system, we can also show that the set of trajectories in the cylindrical phase space is enclosed by the tube structure whose initial set is the closure of the fractal set on the Poincaré section.

First-Kind Cycles of Systems with Cylindrical Phase Space

2020 ◽  
Vol 248 (4) ◽  
pp. 457-466
Author(s):  
S. S. Mamonov ◽  
A. O. Kharlamova
Keyword(s):  
Phase Space ◽  
Cylindrical Phase ◽  
Cylindrical Phase Space

scholarly journals Models for Master-Slave Clock Distribution Networks with Third-Order Phase-Locked Loops

2007 ◽  
Vol 2007 ◽  
pp. 1-17 ◽  
Author(s):  
José Roberto Castilho Piqueira ◽  
Marcela de Carvalho Freschi
Keyword(s):  
Numerical Experiments ◽  
Phase Synchronization ◽  
Distribution Networks ◽  
Phase Locked Loops ◽  
Clock Distribution ◽  
Third Order ◽  
Clock Distribution Networks ◽  
Priority Scheme ◽  
Geographically Distributed ◽  
Lock In

The purpose of this work is to study the processing and transmission of clock signals in networks of geographically distributed nodes, in order to derive conditions for frequency and phase synchronization between the nodes. The focus is on the master-slave architecture, which presents a priority scheme of clock distribution. One-way master-slave (OWMS ) and two-way master-slave (TWMS) chains are studied, considering that the slave nodes are third-order phase-locked loops (PLLs). Third-order PLLs are chosen to improve the transient response but, if their parameters are not well adjusted, stability problems and chaotic behaviors appear, restricting the lock-in range of the network. Lock-in range for third-order PLLs with Sallen-Key filter is determined and it is verified whether this range is reduced when the PLLs are connected to a network. Numerical experiments show how chain size changes the lock-in ranges and the acquisition times.

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