As the distribution of clock signals between the nodes of a network became a critical operational requisite, the transmission delays have to be studied because they affect the accurate recovering of the time basis. In this work, the critical delay value is calculated considering simplifications compatible with practical situations. The main contribution is to consider the dissipative terms in the node equations, expressing critical delays, depending on the time constant of the dissipation.

The phase control problem in a network of oscillators is a problem that appears in several engineering situations: world wide time signal distribution, synchronous communication networks, distributed computation, processes control, and synchronous operation of micro- and nanocircuits. Generally speaking, the distribution of clock signals can be performed controlling the phase of a node oscillation by using the information of the phases of the other nodes that compose the whole network.

Modeling this kind of network presents two types of clock distribution strategies: master-slave (MS) and full-connected (FC), described by Lindsey et al., in a seminal paper that is an important reference for engineers working on design of networks [

Concerning the networks conceived up to the end of the last century, Synchronous Digital Hierarchy (SDH), employing MS strategies, was the most important solution becoming a standard recommended by ITU (International Telecommunication Union) [

Nowadays, it could be considered that clock signal distribution by using MS strategies is well studied and network designers have a lot of good references available for the several types of phase detection and signal filtering [

However, the network architecture evolved to complex topologies with phase and frequency references depending, at each node, on the phase and frequency of almost all the other nodes [

Here, considering that operational frequencies of the node oscillators are normalized, the clock signal phase of each node is supposed to be described by [

The main goal of this paper is to study the several possible behaviors of a network, trying to express them to be related to the connection matrix, the nodes free-running frequencies, and the delays between the nodes. Considering the complexity of the problem, some simplifying assumptions are necessary in order to treat the problem.

The first simplification is about the network structure that is supposed to be triangular [

In the next section, the equations for this type of configuration will be derived with the spatial phase errors being the measurable state variables. Then, in order to analyze the tracking mode of the system, a linear model is developed, resulting in two decoupled equations relating the spatial errors to the delayed spatial errors.

Based on these equations, it is possible to calculate the critical delay that implies instability and to relate it to time constant

Following the reasoning and notation exposed in the former section, a three-node PLL network can be modeled by the following equations:

At this point, it is assumed that the nodes have the same constitutive parameters, that is,

In order to be more accurate from an engineering point of view, the delays between the nodes ought to be different. But in real modern networks, they assume low values, being fractions about

Taking the former conditions into consideration, the equations of the phase nodes become

To proceed with the analysis of the network, it must be considered that to obtain the node clocks in a correct sequence, the phase and frequency errors between the nodes have to vanish after any kind of perturbation, with as minimum overshoot and tracking time as possible. Consequently, the important state variables are the spatial phase (

In this case, a three-node network, there are six index combinations

An important indicator of the performance of a clock distribution systems is regarding its behavior in the tracking mode; that is, how the system recovers the synchronism when a perturbation around the synchronous state occurs [

In this case, the phase deviation can be considered to be small and the argument of the

Consequently, the model for the tracking mode of a triangular PLL network is given by two decoupled equations, which are ordinary, homogeneous, and linear, containing a delay term. This equation admits an analytical solution. According to [

The critical delay value for the three-node clock distribution network can be determined considering that the spatial errors are described by

In order to determine how the bifurcation parameter

For the dynamical system described by (

It is supposed that there exists

Consequently, the module condition

The number of values for

Then, from (

By computing the module condition given by (

Equation (

It can be noticed that there are some situations with (

For a system given by (

Figure

Real positive values for

However, considering a practical point of view, it is important to know how the critical delay values (

The results presented here enable the design of a clock distribution network with second-order dissipative nodes [

The general idea is to maintain the delays of the network smaller than the critical value derived here. Under this condition, the reachability of the synchronous state follows the criteria shown in [

As the equations considered here are for normalized angular frequency and the time constant

Consequently, taking

Simulations of (

Simulations for (

Finally, it can be concluded that

The authors declare that there is no conflict of interests regarding the publication of this paper.