Linear Approach for Synchronous State Stability in Fully Connected PLL Networks

Abstract

Synchronization is an essential feature for the use of digital systems in telecommunication networks, integrated circuits, and manufacturing automation. Formerly, master-slave (MS) architectures, with precise master clock generators sending signals to phase-locked loops (PLLs) working as slave oscillators, were considered the best solution. Nowadays, the development of wireless networks with dynamical connectivity and the increase of the size and the operation frequency of integrated circuits suggest that the distribution of clock signals could be more efficient if distributed solutions with fully connected oscillators are used. Here, fully connected networks with second-order PLLs as nodes are considered. In previous work, how the synchronous state frequency for this type of network depends on the node parameters and delays was studied and an expression for the long-term frequency was derived (Piqueira, 2006). Here, by taking the first term of the Taylor series expansion for the dynamical system description, it is shown that for a generic network with 𝑁 nodes, the synchronous state is locally asymptotically stable.

1. Introduction

Digital engineering technologies for communications, control, and computation require reliable clock distribution systems to guarantee the correct temporal order in the information processing by the several parts of a spatially distributed system [1–4]. Synchronization network is the general denomination of the part of the whole system responsible for this temporal order and the several possible solutions for its design are presented in [5].

Originally, master-slave architectures were used to distribute a precise clock signal generated by a master node to the other points of the systems where PLLs regenerate the phase and frequency information [1, 5].

The evolution of the telecommunication services to wireless and dynamical networks has shown the inadequacy of centralized clock distribution structures in these cases, motivating the study of fully connected architectures to generate reference signals with the phase-locked loops operating as nodes of the synchronization networks [1, 5–7].

Besides telecommunication networks, the main fields for the application of fully connected systems are time signal distribution in digital electronic circuits [2, 7–9] and wireless sensor networks [10]. Another very important application of networks is the implementation of oscillatory neural-computing devices, where the vector of phase-differences amongst a group of synchronized oscillators is associated with some memory information [11, 12].

In this work, a fully connected 𝑁-node PLL network is studied, starting with a review of the model for a single node and the derivation of the synchronous state frequency [13]. Then, the dynamical equation for the phase differences between nodes is presented providing a model for the whole 𝑁-node network.

As it was shown in [13, 14], this kind of network presents a synchronous state that is reachable for any possible combination of node parameters. Consequently, the linear approximation gives practical hints about the lock-in operation of the network [15]. Then, considering the first term of the Taylor series development around the synchronous state [16], it is shown that the synchronous state is locally asymptotically stable for any number 𝑁 of nodes.

2. Nodes in a Fully Connected Architecture and Synchronous State Frequency

In this section, the nodes of the fully connected network are analyzed, considering that they are second-order phase-locked loops modeled as the classical analog version [17]. The results shown in [13] are summarized and the expression of synchronous state frequency is derived. This expression will be used to determine the stability conditions. PLL nodes are closed loops composed of a phase detector (PD), a low-pass filter (F), and a voltage-controlled oscillator (VCO) [17]. The double frequency jitter in the output of the phase detector is neglected [18–20].

The 𝑁 nodes are geographically distributed and mutually coupled by using two-way connections. Each 𝑖-node exchanges clock signals with all the 𝑗-nodes such that 𝑗≠𝑖. Each VCO belonging to a PLL is described by its free-running frequency 𝜔_𝑖 and by its phase 𝜙_𝑖(𝑡). The whole network has its dynamics described by phase errors and frequency errors defined by Δ𝜙_𝑗𝑖(𝑡) and Δ𝜙̇‌_𝑗𝑖(𝑡), respectively. Time delays 𝜏_𝑗𝑖 corresponding to the propagation of signals from the node 𝑗 VCO output to the node 𝑖 PD input are considered, as shown in Figure 1, containing the model of the signal processed in node 1.

Figure 1 

Node 1 signal processing in a fully connected PLL network.

It can be noticed that in an 𝑁-node fully connected PLL network, the nodes need 𝑁−1 phase detectors [21] where the local VCO output signal is multiplied by the delayed signals sent by the VCO of the other nodes. As there is no integrated circuit with this architecture, the implementation of the nodes requires a combination of several PLL chips with the outputs of their phase detectors weighted and the result being the input of a single filter that feeds a single VCO. For the node 1 from Figure 1, the definitions are as follows: (i)𝜏₂₁: delay from node 2 to node 1; (ii)𝜏₃₁: delay from node 3 to node 1;(iii)𝜏_𝑁1: delay from node 𝑁 to node 1; (iv)𝑣₁(𝑡): VCO output of the node 1;(v)𝑣₂(𝑡−𝜏₂₁): VCO output of the node 2 with a delay 𝜏₂₁;(vi)𝑣₃(𝑡−𝜏₃₁): VCO output of the node 3 with a delay 𝜏₃₁;(vii)𝑣_𝑁(𝑡−𝜏_𝑁1): VCO output of the node 𝑁 with a delay 𝜏_𝑁1;(viii)𝑣_𝑑21(𝑡): output of PD 2;(ix)𝑣_𝑑31(𝑡): output of PD 3;(x)𝑣_𝑑𝑁1(𝑡): output of PD 𝑁;(xi)𝑣_𝑐1(𝑡): output of F that controls the VCO.

The instantaneous individual VCO phases and the instantaneous phase errors, 𝜙_𝑖(𝑡) and Δ𝜙_𝑗𝑖(𝑡), are expressed in the forms 𝜙_𝑖(𝑡)=𝜔_𝑖𝑡+𝜃_𝑖(𝑡),Δ𝜙_𝑗𝑖(𝑡)=𝜙_𝑗(𝑡)−𝜙_𝑖(𝑡).(2.1)

The node parameters related to PLL operation can be characterized by the following:(i)PD multiplying factors: 𝑘_𝑚1=𝑘_𝑚2=⋯=𝑘_𝑚𝑖=𝑘_𝑚, in volts⁻¹, with 𝑖=1,2,…,𝑁;(ii)gains of the VCOs: 𝑘₁=𝑘₂=⋯=𝑘_𝑖=𝑘₀, in rad/sV, with 𝑖=1,2,…,𝑁;(iii) cut-off frequencies of F: 𝜇₁₁=𝜇₁₂=𝜇_1𝑖=⋯=𝜇₁, in rad/s, with 𝑖=1,2,…,𝑁. The constitutive parameters of the nodes are considered to be the same, in order to simplify the analytical reasoning. If the filters cut-off frequencies are different, but sufficient to avoid double frequency jitter [18], the results derived here are basically the same and only the acquisition times change [13]. If VCO and PD gains are considered to be different, the problem has to be numerically treated, and the expression of the synchronous state frequency derived here changes [22].

Under the assumption that all nodes have the same constitutive parameters, the output of each VCO is𝑣_𝑖(𝑡)=𝑉cos𝜙_𝑖(𝑡);(2.2)and signals received by the phase detector of node 𝑖 from node 𝑗, with propagation delays 𝜏_𝑗𝑖, can be written as𝑣_𝑗𝑡−𝜏_𝑗𝑖=𝑉sin𝜙_𝑗𝑡−𝜏_𝑗𝑖,(2.3)where 𝑉 is the controlled amplitude of the outputs of VCOs and PDs. Considering that each phase detector 𝑗≠𝑖, belonging to node 𝑖, is a signal multiplier [17]:𝑣_𝑑𝑗𝑖(𝑡)=𝑘_𝑚𝑣_𝑗𝑡−𝜏_𝑗𝑖𝑣_𝑖(𝑡),(2.4)replacing (2.2) and (2.3) in (2.4), and neglecting the double frequency components [18–20] as their frequency are much greater than the filter cut-off frequency, the output of each phase detector 𝑗≠𝑖, belonging to node 𝑖, is given by𝑣_𝑑𝑗𝑖(𝑡)=𝑘_𝑚𝑉²sin𝜙_𝑗𝑡−𝜏_𝑗𝑖cos𝜙_𝑖(𝑡).(2.5)Each resulting signal given by (2.5) is multiplied by 1/(𝑁−1) and added, in order to compose the filter input as below:𝑣_𝑑𝑖(𝑡)=1𝑁−1𝑘_𝑚𝑉²2{^𝑁_{𝑗=1,𝑗≠𝑖}sin𝜙_𝑗𝑡−𝜏_𝑗𝑖−𝜙_𝑖(𝑡)}.(2.6)Defining 𝑘_𝑑=(1/2)(𝑘_𝑚𝑉²), expression (2.6) is simplified to𝑣_𝑑𝑖(𝑡)=𝑘_𝑑𝑁−1{^𝑁_{𝑗=1,𝑗≠𝑖}sin𝜙_𝑗𝑡−𝜏_𝑗𝑖−𝜙_𝑖(𝑡)}.(2.7)

The filters are considered to be first-order low-pass implying second-order nodes. This choice is a common practice because second-order PLLs always reach a synchronous state when submitted to phase steps and ramps [17] in spite of the double-frequency jitter [19, 20]. If more accurate transient responses are necessary, second-order filters are used, but complicated behaviors like bifurcation and chaos appear [23], worsening the operation.

Consequently, equations for the dynamics of the VCO phase are obtained by considering the filter transfer function 𝐹_𝑖(𝑠)=𝜇₁/(𝑠+𝜇₁) [15], resulting in the expression𝑣̇‌_𝑐𝑖(𝑡)+𝜇₁𝑣_𝑐𝑖(𝑡)=𝜇₁𝑣_𝑑𝑖(𝑡).(2.8)

Replacing 𝑣_𝑑𝑖, given by (2.7), and the VCO control signal 𝑣_𝑐𝑖=𝜃̇‌_𝑖(𝑡)/𝑘₀ in (2.8), the equation for the node phase is𝜃̈‌_𝑖(𝑡)+𝜇₁𝜃̇‌_𝑖(𝑡)−𝜇₁𝑘_𝑜𝑘_𝑑(𝑁−1){^𝑁_{𝑗=1,𝑗≠𝑖}sin𝜙_𝑗𝑡−𝜏_𝑗𝑖−𝜙_𝑖(𝑡)}=0.(2.9)Then, considering 𝜙_𝑖(𝑡)=𝑤_𝑖𝑡+𝜃_𝑖(𝑡) in (2.9) and defining 𝜇₂=𝑘_𝑜𝑘_𝑑 and 𝑘=𝜇₁𝜇₂/(𝑁−1), the dynamics of each VCO phase in a fully connected network is given by𝜙̈‌_𝑖(𝑡)+𝜇₁𝜙̇‌_𝑖(𝑡)−𝜇₁𝜔_𝑖−𝑘{^𝑁_𝑗=_{1,𝑗≠𝑖}sin𝜙_𝑗𝑡−𝜏_𝑗𝑖−𝜙_𝑖(𝑡)}=0.(2.10)Equation (2.10) is similar to the pendulum equation, containing a dissipation component 𝜇₁𝜙̇‌_𝑖(𝑡), a delayed conservative term 𝑘∑^𝑁_{𝑗=1,𝑗≠𝑖}sin[𝜙_𝑗(𝑡−𝜏_𝑗𝑖)−𝜙_𝑖(𝑡)], and a forcing part 𝜇₁𝜔_𝑖 [16]. Consequently, it is reasonable to suppose that the long-term solution of the system is a synchronous state, as shown in [14, 18], with the phases of all nodes oscillating with the same frequency 𝜔_𝑠 that can be estimated. Thus, in order to estimate this frequency, the following hypotheses are considered [13]:(a)𝜙̇‌_𝑖(𝑡)=𝜔_𝑠,(b)𝜙̈‌_𝑖(𝑡)=0,(c)𝜙_𝑗(𝑡−𝜏_𝑗𝑖)≈𝜙_𝑗(𝑡)+𝜙̇‌_𝑖(𝑡)(−𝜏_𝑗𝑖)≈𝜙_𝑗(𝑡)−𝜔_𝑠𝜏_𝑗𝑖.

Therefore,𝜇₁𝜔_𝑠−𝜔_𝑖−𝑘[^𝑁_{𝑗=1,𝑗≠𝑖}sinΔ𝜙_𝑗𝑖−𝜔_𝑠𝜏_𝑗𝑖]=0.(2.11)The values of (Δ𝜙_𝑗𝑖−𝜔_𝑠𝜏_𝑗𝑖) are considered to be small because in the majority of the practical situations, the network is operating in the lock-in mode [17]. Consequently, (2.11) can be written as a linear approximation considering sin[Δ𝜙_𝑗𝑖−𝜔_𝑠𝜏_𝑗𝑖]≈Δ𝜙_𝑗𝑖−𝜔_𝑠𝜏_𝑗𝑖. Hence, for each node 𝑖,𝜇₁𝜔_𝑠−𝜇₁𝜔_𝑖−𝑘[^𝑁_{𝑗=1,𝑗≠𝑖}Δ𝜙_𝑗𝑖]+𝑘𝜔_𝑠[^𝑁_{𝑗=1,𝑗≠𝑖}𝜏_𝑗𝑖]=0.(2.12)Using (2.12) for an 𝑁-node network, with 𝑖,𝑗=1,…,𝑁, and 𝑗≠𝑖, as well as adding the 𝑁 resulting equations, as the sum of the terms Δ𝜙_𝑗𝑖 is equal to zero because Δ𝜙_𝑗𝑖=−Δ𝜙_𝑖𝑗, one can write𝑁𝜇₁𝜔_𝑠−𝜇^1𝑁_𝑖=1𝜔_𝑖+𝑘𝜔_𝑠(^𝑁_𝑖=1^𝑁_{𝑗=1,𝑗≠𝑖}𝜏_𝑗𝑖)=0.(2.13)Calculating 𝜔_𝑠 from (2.13),𝜔_𝑠=𝜇₁(∑^𝑁_𝑖=1𝜔_𝑖)𝑁𝜇₁+𝑘(∑^𝑁_𝑖=1∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝜏_𝑗𝑖).(2.14)Dividing (2.14) by 𝑁𝜇₁, and replacing 𝑘=𝜇₁𝜇₂/(𝑁−1), the estimation of the synchronous state frequency (𝜔_𝑠) is obtained:𝜔_𝑠=(1/𝑁)(∑^𝑁_𝑖=1𝜔_𝑖)1+(𝜇₂/𝑁(𝑁−1))(∑^𝑁_𝑖=1∑^𝑁_{𝑗=𝑖,𝑗≠𝑖}𝜏_𝑗𝑖).(2.15)Therefore, expression (2.15) is an estimation for the frequency of the synchronous state for a fully connected second-order PLL network, depending on the individual free-running frequencies and propagation delays. Notice that when the delays are zero, 𝜔_𝑠 is given by the mean value of 𝜔_𝑖. As it was considered that all free-running frequencies are different, there are phase shifts between the nodes in the synchronous state. In practical cases, as in communication networks, these phase differences are object of delay compensation techniques [5].

In previous work [13], numerical simulations were conducted to investigate the accuracy of expression (2.15) and to analyze how gains and delays change the behavior of the network. Here, the dynamic equations for the phase differences are derived allowing the analytic study of the local stability of the synchronous state.

3. Phase Difference Equations

In this section, the equations describing the dynamics of the phase errors, Δ𝜙_𝑗𝑖(𝑡), are derived. A set of 𝑁−1 second-order ordinary differential equations is obtained, expressing the phase differences between all the nodes and the node 1, taken as reference.

Starting with the individual dynamical equations for each VCO phase and expressing the differences between the nodes 2,3,…,𝑁 and node 1, equations for Δ𝜙_𝑗1(𝑡), 𝑗=2,3,…,𝑁 are written as below: VCO1:𝜙̈‌₁(𝑡)+𝜇₁𝜙̇‌₁(𝑡)−𝜇₁𝜔₁−𝑘×sin𝜙₂𝑡−𝜏₂₁−𝜙₁(𝑡)+sin𝜙₃𝑡−𝜏₃₁−𝜙₁(𝑡)+sin𝜙₄𝑡−𝜏₄₁−𝜙₁(𝑡)]+⋯+sin𝜙_𝑁𝑡−𝜏_𝑁1−𝜙₁(𝑡)=0;VCO2:𝜙̈‌₂(𝑡)+𝜇₁𝜙̇‌₂(𝑡)−𝜇₁𝜔₂−𝑘×sin𝜙₁𝑡−𝜏₁₂−𝜙₂(𝑡)+sin𝜙₃𝑡−𝜏₃₂−𝜙₂(𝑡)+sin𝜙₄𝑡−𝜏₄₂−𝜙₂(𝑡)+⋯+sin𝜙_𝑁𝑡−𝜏_𝑁2−𝜙₂(𝑡)=0;VCO3:𝜙̈‌₃(𝑡)+𝜇₁𝜙̇‌₃(𝑡)−𝜇₁𝜔₃−𝑘×sin𝜙₁𝑡−𝜏₁₃−𝜙₃(𝑡)+sin𝜙₂𝑡−𝜏₃₂−𝜙₃(𝑡)+sin𝜙₄𝑡−𝜏₄₃−𝜙₃(𝑡)+⋯+sin𝜙_𝑁𝑡−𝜏_𝑁3−𝜙₃(𝑡)=0;⋮VCO𝑁−1:𝜙̈‌_𝑁−1(𝑡)+𝜇₁𝜙̇‌_𝑁−1(𝑡)−𝜇₁𝜔_𝑁−1−𝑘×sin𝜙₁𝑡−𝜏_1𝑁−1−𝜙_𝑁−1(𝑡)+sin𝜙₂𝑡−𝜏_2𝑁−1−𝜙_𝑁−1(𝑡)+sin𝜙₃𝑡−𝜏_3𝑁−1−𝜙_𝑁−1(𝑡)+⋯+sin𝜙_𝑁𝑡−𝜏_𝑁−1𝑁−𝜙_𝑁−1(𝑡)=0;VCO𝑁:𝜙̈‌_𝑁(𝑡)+𝜇₁𝜙̇‌_𝑁(𝑡)−𝜇₁𝜔_𝑁−𝑘×sin𝜙₁𝑡−𝜏_1𝑁−𝜙_𝑁(𝑡)+sin𝜙₂𝑡−𝜏_2𝑁−𝜙_𝑁(𝑡)+sin𝜙₃𝑡−𝜏_3𝑁−𝜙_𝑁(𝑡)]+⋯+sin𝜙_𝑁−1𝑡−𝜏_𝑁−1𝑁−𝜙_𝑁(𝑡)=0.(3.1) Using the condition 𝜙_𝑖(𝑡−𝜏_𝑗𝑖)≈𝜙_𝑖(𝑡)−𝜔_𝑠𝜏_𝑗𝑖 and considering Δ𝜙_𝑗1(𝑡)=𝜙_𝑗(𝑡)−𝜙₁(𝑡), then VCO1:𝜙̈‌₁(𝑡)+𝜇₁𝜙̇‌₁−𝜇₁𝜔₁−𝑘×sinΔ𝜙₂₁(𝑡)−𝜔_𝑠𝜏₂₁+sinΔ𝜙₃₁(𝑡)−𝜔_𝑠𝜏₃₁+sinΔ𝜙₄₁(𝑡)−𝜔_𝑠𝜏₄₁+⋯+sinΔ𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0;VCO2:𝜙̈‌₂(𝑡)+𝜇₁𝜙̇‌₂(𝑡)−𝜇₁𝜔₂−𝑘×sinΔ𝜙₁₂(𝑡)−𝜔_𝑠𝜏₁₂+sinΔ𝜙₃₂(𝑡)−𝜔_𝑠𝜏₃₂+sinΔ𝜙₄₂(𝑡)−𝜔_𝑠𝜏₄₂+⋯+sinΔ𝜙_𝑁2(𝑡)−𝜔_𝑠𝜏_𝑁2=0;VCO3:𝜙̈‌₃(𝑡)+𝜇₁𝜙̇‌₃(𝑡)−𝜇₁𝜔₃−𝑘×sinΔ𝜙₁₃(𝑡)−𝜔_𝑠𝜏₁₃+sinΔ𝜙₂₃(𝑡)−𝜔_𝑠𝜏₂₃+sinΔ𝜙₄₃(𝑡)−𝜔_𝑠𝜏₄₃+⋯+sinΔ𝜙_𝑁3(𝑡)−𝜔_𝑠𝜏_𝑁3=0;⋮VCO𝑁−1:𝜙̈‌_𝑁−1(𝑡)+𝜇₁𝜙̇‌_𝑁−1(𝑡)−𝜇₁𝜔_𝑁−1−𝑘×sinΔ𝜙_1𝑁−1(𝑡)−𝜔_𝑠𝜏_1𝑁−1+sinΔ𝜙_2𝑁−1(𝑡)−𝜔_𝑠𝜏_2𝑁−1+sinΔ𝜙_3𝑁−1(𝑡)−𝜔_𝑠𝜏_3𝑁−1+⋯+sinΔ𝜙_𝑁𝑁−1(𝑡)−𝜔_𝑠𝜏_𝑁𝑁−1=0;VCO𝑁:𝜙̈‌_𝑁(𝑡)+𝜇₁𝜙̇‌_𝑁(𝑡)−𝜇₁𝜔_𝑁−𝑘×sinΔ𝜙_1𝑁(𝑡)−𝜔_𝑠𝜏_1𝑁+sinΔ𝜙_2𝑁(𝑡)−𝜔_𝑠𝜏_2𝑁+sinΔ𝜙_3𝑁(𝑡)−𝜔_𝑠𝜏_3𝑁+⋯+sinΔ𝜙_𝑁−1𝑁(𝑡)−𝜔_𝑠𝜏_𝑁−1𝑁=0.(3.2) Expressing the differences with node 1 as reference, VCOs1and2:𝜙̈‌₂(𝑡)−𝜙̈‌₁(𝑡)+𝜇₁𝜙̇‌₂(𝑡)−𝜙̇‌₁(𝑡)−𝜇₁𝜔₂−𝜔₁−𝑘sinΔ𝜙₁₂(𝑡)−𝜔_𝑠𝜏₁₂−𝑘sinΔ𝜙₃₂(𝑡)+𝜔_𝑠𝜏₃₂−𝑘sinΔ𝜙₄₂(𝑡)−𝜔₂𝜏₄₂−⋯−𝑘sinΔ𝜙_𝑁2(𝑡)−𝜔_𝑠𝜏_𝑁2+𝑘sinΔ𝜙₂₁(𝑡)−𝜔_𝑠𝜏₂₁+⋯+𝑘sinΔ𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0;VCOs1and3:𝜙̈‌₃(𝑡)−𝜙̈‌₁(𝑡)+𝜇₁𝜙̇‌₃(𝑡)−𝜙̇‌₁(𝑡)−𝜇₁𝜔₃−𝜔₁−𝑘sinΔ𝜙₁₃(𝑡)−𝜔_𝑠𝜏₁₃−𝑘sinΔ𝜙₂₃(𝑡)−𝜔_𝑠𝜏₂₃−𝑘sinΔ𝜙₄₃(𝑡)−𝜔₂𝜏₄₃−⋯−𝑘sinΔ𝜙_𝑁3(𝑡)−𝜔_𝑠𝜏_𝑁3+𝑘sinΔ𝜙₂₁(𝑡)+𝜔_𝑠𝜏₂₁+𝑘sinΔ𝜙₃₁(𝑡)+𝜔_𝑠𝜏₃₁+𝑘sinΔ𝜙₄₁(𝑡)+𝜔_𝑠𝜏₄₁+⋯+𝑘sin𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0;⋮VCOs1and𝑁:𝜙̈‌_𝑁(𝑡)−𝜙̈‌₁(𝑡)+𝜇₁𝜙̇‌_𝑁(𝑡)−𝜙̇‌₁(𝑡)−𝜇₁𝜔_𝑁−𝜔₁−𝑘sinΔ𝜙_1𝑁(𝑡)−𝜔_𝑠𝜏_1𝑁−𝑘sinΔ𝜙_2𝑁(𝑡)−𝜔_𝑠𝜏_2𝑁−𝑘sinΔ𝜙_3𝑁(𝑡)−𝜔₂𝜏_3𝑁−⋯−𝑘sinΔ𝜙_𝑁−1𝑁(𝑡)−𝜔_𝑠𝜏_𝑁−1𝑁+𝑘sinΔ𝜙₂₁(𝑡)+𝜔_𝑠𝜏₂₁+𝑘sinΔ𝜙₃₁(𝑡)+𝜔_𝑠𝜏₃₁+𝑘sinΔ𝜙₄₁(𝑡)+𝜔_𝑠𝜏₅₂+⋯+𝑘sin𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0.(3.3) Replacing the terms Δ𝜙_𝑗1(𝑡)=𝜙_𝑗(𝑡)−𝜙₁(𝑡), Δ𝜙̇‌_𝑗1(𝑡)=𝜙̇‌_𝑗(𝑡)−𝜙̇‌₁(𝑡), Δ𝜙̈‌_𝑗1(𝑡)=𝜙̈‌_𝑗(𝑡)−𝜙̈‌₁(𝑡), and using the identity sin[Δ𝜙_𝑗1(𝑡)−𝜔_𝑠𝜏_𝑗1]+sin[Δ𝜙_𝑗𝑖(𝑡)+𝜔_𝑠𝜏_𝑗𝑖]=2sinΔ𝜙_𝑗𝑖cos𝜔_𝑠𝜏_𝑗𝑖 in the former expressions, the VCO phase differences become VCOs1and2:Δ𝜙̈‌₂₁(𝑡)+𝜇₁Δ𝜙̇‌₂₁(𝑡)−𝜇₁Δ𝜔₂₁+2𝑘sinΔ𝜙₂₁(𝑡)cos𝜔_𝑠𝜏₂₁−𝑘sinΔ𝜙₃₂(𝑡)+𝜔_𝑠𝜏₃₂−𝑘sinΔ𝜙₄₂(𝑡)−𝜔₂𝜏₄₂−⋯−𝑘sinΔ𝜙_𝑁2(𝑡)−𝜔_𝑠𝜏_𝑁2+𝑘sinΔ𝜙₃₁(𝑡)−𝜔_𝑠𝜏₃₁+𝑘sinΔ𝜙₄₁(𝑡)−𝜔_𝑠𝜏₄₁+⋯+𝑘sinΔ𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0;VCOs1and3:Δ𝜙̈‌₃₁(𝑡)+𝜇₁Δ𝜙̇‌₃₁(𝑡)−𝜇₁Δ𝜔₃₁+2𝑘sinΔ𝜙₃₁(𝑡)cos𝜔_𝑠𝜏₃₁−𝑘sinΔ𝜙₂₃(𝑡)+𝜔_𝑠𝜏₂₃−𝑘sinΔ𝜙₄₃(𝑡)−𝜔₂𝜏₄₃−⋯−𝑘sinΔ𝜙_𝑁3(𝑡)−𝜔_𝑠𝜏_𝑁3+𝑘sinΔ𝜙₂₁(𝑡)−𝜔_𝑠𝜏₂₁+𝑘sinΔ𝜙₃₁(𝑡)−𝜔_𝑠𝜏₃₁+⋯+𝑘sinΔ𝜙_𝑁1(𝑡)−𝜔_𝑠𝜏_𝑁1=0;⋮VCOs1and𝑁:Δ𝜙̈‌_𝑁1(𝑡)+𝜇₁Δ𝜙̇‌_𝑁1(𝑡)−𝜇₁Δ𝜔_𝑁1+2𝑘sinΔ𝜙_𝑁1(𝑡)cos𝜔_𝑠𝜏_𝑁1−𝑘sinΔ𝜙_2𝑁(𝑡)+𝜔_𝑠𝜏_2𝑁−𝑘sinΔ𝜙_3𝑁(𝑡)−𝜔₂𝜏_3𝑁−⋯−𝑘sinΔ𝜙_𝑁−1𝑁(𝑡)−𝜔_𝑠𝜏_𝑁−1𝑁+𝑘sinΔ𝜙₂₁(𝑡)−𝜔_𝑠𝜏₂₁+𝑘sinΔ𝜙₃₁(𝑡)−𝜔_𝑠𝜏₃₁+⋯+𝑘sinΔ𝜙_𝑁−11(𝑡)−𝜔_𝑠𝜏_𝑁−11=0.(3.4) These equations give the general expression that describes the dynamical behavior of the phase differences for the network VCOs as below:Δ𝜙̈‌_𝑗1(𝑡)+𝜇₁Δ𝜙̇‌_𝑗1(𝑡)−𝜇₁Δ𝜔_𝑗1+2𝑘sinΔ𝜙_𝑗1(𝑡)cos𝜔_𝑠𝜏_𝑗1−𝑘{^𝑁_{𝑚=2,𝑚≠𝑗}sinΔ𝜙_𝑚𝑗(𝑡)−𝜔_𝑠𝜏_𝑚𝑗}+𝑘{^𝑁_{𝑚=2,𝑚≠𝑗}sinΔ𝜙_𝑚1(𝑡)−𝜔_𝑠𝜏_𝑚1}=0.(3.5)The nonlinear differential equation (3.5), for small-phase deviations, can be approximately expressed by the linear term of the Taylor series expansion [16]. Then, rewriting (3.5)Δ𝜙̈‌_𝑗1(𝑡)+𝜇₁Δ𝜙̇‌_𝑗1(𝑡)−𝜇₁Δ𝜔_𝑗1+2𝑘Δ𝜙_𝑗1(𝑡)−𝑘{^𝑁_{𝑚=2,𝑚≠𝑗}Δ𝜙_𝑚𝑗(𝑡)−Δ𝜙_𝑚1(𝑡)}+𝑘𝜔_𝑠{^𝑁_{𝑚=1,𝑚≠𝑗}𝜏_𝑚𝑗−𝜏_𝑚1}=0.(3.6)Expressions (3.5) and (3.6) describe the dynamics of the phase adjustments of a fully connected PLL network depending on the PLL node parameters 𝜇₁ and 𝑘, the number of nodes 𝑁, as well as the individual free-running frequencies and the delays. These equations allow the research of the synchronous state stability that is conducted in the next section.

4. Synchronous State Stability

In Section 2, an expression for the synchronous state frequency for the fully connected network was derived and, in this section, the stability of the synchronous state is studied, that is, if the reachable synchronous state is robust under small perturbations. The analysis is performed considering that the solutions of the linear equation (3.6) can be topologically equivalent to the solutions of (3.5) in a small neighborhood of the synchronous state if it is a hyperbolic equilibrium point [16]. The procedure will be shown for three-node and four-node networks providing the identification of patterns in the expressions for the eigenvalues of the Jacobian matrix representing the linear equivalent system around the synchronous state.

4.1. Three-Node Network

For a three-node network, the phase differences are Δ𝜙₁₂ and Δ𝜙₁₃. By using (3.5), the system is described by VCOs1and2:Δ𝜙̈‌₁₂+𝜇₁Δ𝜙̇‌₁₂+2𝑘Δ𝜙₁₂+𝑘Δ𝜙₁₃−𝑘Δ𝜙₂₃+𝜇₁Δ𝜔₂₁+𝑘𝜔_𝑠𝜏₁₃−𝜏₂₃_{𝑎}=0;VCOs1and3:Δ𝜙̈‌₁₃+𝜇₁Δ𝜙̇‌₁₃+2𝑘Δ𝜙₁₃+𝑘Δ𝜙₁₂+𝑘Δ𝜙₂₃+𝜇₁Δ𝜔₃₁+𝑘𝜔_𝑠𝜏₁₂−𝜏₃₂_{𝑏}=0.(4.1) The system composed of two second-order equations (4.1) can be described by the state variables:𝑥₁≡Δ𝜙₁₂,𝑥₃≡Δ𝜙₁₃,𝑥₂≡Δ𝜙̇‌₁₂,𝑥₄≡Δ𝜙̇‌₁₃.(4.2)Considering that Δ𝜙₂₃=𝑥₃−𝑥₁, the following first-order equations result:𝑥̇₁=𝑥₂,𝑥̇₂=−3𝑘𝑥₁−𝜇₁𝑥₂−𝑎,𝑥̇₃=𝑥₄,𝑥̇₄=−3𝑘𝑥₃−𝜇₁𝑥₄−𝑏.(4.3)

Consequently, the Jacobian matrix 𝐽 from (4.3) is𝐽=⎡⎢⎢⎢⎢⎢⎣0100−3𝑘−𝜇₁00000100−3𝑘−𝜇₁⎤⎥⎥⎥⎥⎥⎦.(4.4)The eigenvalues of 𝐽 are the roots of the characteristic polynomial 𝜆²+𝜇₁𝜆+3𝑘²=0,(4.5)which have multiplicity two, and are given by𝜆₁=−𝜇₁2+𝜇²₁−12𝑘2,𝜆₂=−𝜇₁2−𝜇²₁−12𝑘2.(4.6)

4.2. Four-Node Network

The calculation of the eigenvalues for the four-node network follows the same procedure of the former case. The phase differences are represented by Δ𝜙₁₂, Δ𝜙₁₃, and Δ𝜙₁₄. By using (3.5), the system is written as VCOs1and2:Δ𝜙̈‌₁₂+𝜇₁Δ𝜙̇‌₁₂+2𝑘Δ𝜙₁₂+𝑘Δ𝜙₁₃−𝑘Δ𝜙₂₃+𝑘Δ𝜙₁₄−𝑘Δ𝜙₂₄+𝜇₁Δ𝜔₂₁+𝑘𝜔_𝑠𝜏₁₃−𝜏₂₃+𝜏₁₄−𝜏₂₄_{𝑎}=0;VCOs1and3:Δ𝜙̈‌₁₃+𝜇₁Δ𝜙̇‌₁₃+2𝑘Δ𝜙₁₃+𝑘Δ𝜙₁₂+𝑘Δ𝜙₂₃+𝑘Δ𝜙₁₄−𝑘Δ𝜙₃₄+𝜇₁Δ𝜔₃₁+𝑘𝜔_𝑠𝜏₁₂−𝜏₃₂+𝜏₁₄−𝜏₃₄_{𝑏}=0;VCOs1and4:Δ𝜙̈‌₁₄+𝜇₁Δ𝜙̇‌₁₄+2𝑘Δ𝜙₁₄+𝑘Δ𝜙₁₂+𝑘Δ𝜙₂₄+𝑘Δ𝜙₁₃+𝑘Δ𝜙₃₄+𝜇₁Δ𝜔₄₁+𝑘𝜔_𝑠𝜏₁₂−𝜏₄₂+𝜏₁₃−𝜏₄₃_{𝑐}=0.(4.7) The second-order expressions (4.7) can be rewritten in terms of state variables:𝑥₁≡Δ𝜙₁₂,𝑥₃≡Δ𝜙₁₃,𝑥₅≡Δ𝜙₁₄,𝑥₂≡Δ𝜙̇‌₁₂,𝑥₄≡Δ𝜙̇‌₁₃,𝑥₆≡Δ𝜙̇‌₁₄.(4.8)Considering that Δ𝜙₂₃=𝑥₃−𝑥₁, Δ𝜙₂₄=𝑥₅−𝑥₁, and Δ𝜙₃₄=𝑥₃−𝑥₅, the following first-order equations result:𝑥̇₁=𝑥₂,𝑥̇₂=−4𝑘𝑥₁−𝜇₁𝑥₂−𝑎,𝑥̇₃=𝑥₄,𝑥̇₄=−4𝑘𝑥₃−𝜇₁𝑥₄−𝑏,𝑥̇₅=𝑥₆,𝑥̇₆=−4𝑘𝑥₅−𝜇₁𝑥₆−𝑐.(4.9)

Consequently, the Jacobian matrix 𝐽 from (4.9) is𝐽=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣010000−4𝑘−𝜇₁000000010000−4𝑘−𝜇₁000000010000−4𝑘−𝜇₁⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦.(4.10)The eigenvalues of 𝐽 are given by the roots of the characteristic polynomial: 𝜆²+𝜇₁𝜆+4𝑘³=0,(4.11)which have multiplicity 3, and are𝜆₁=−𝜇₁2+𝜇²₁−16𝑘2,𝜆₂=−𝜇₁2−𝜇²₁−16𝑘2.(4.12)

4.3. N-Node Network

As it was shown, for a fully connected PLL network, the Taylor series development around the synchronous state results in Jacobian matrices with a canonical Jordan form [24]. Table 1 shows the characteristic polynomials and the expressions of the eigenvalues and their multiplicities, m, corresponding to fully connected networks dynamic equations, around the synchronous state, for different number, 𝑛, of nodes.

Table 1 

Eigenvalues for an 𝑁-node network.

It can be observed that for any number of nodes, the eigenvalues from Table 1 are with negative real parts, implying that the synchronous state is locally asymptotically stable.

5. Conclusions

As the fully connected architecture started to be used in large scale in clock distribution systems, (2.15) can be applied by network designers as an estimation for the frequency of the synchronous state when second-order PLLs are used to extract the timing information in the nodes.

Besides, because the synchronous state is a hyperbolic locally asymptotically stable equilibrium point, the network can recover synchronization after perturbation. The eigenvalues presented in Table 1 can be used for determining the transient behavior of such a resynchronization process.

Acknowledgment

José R. C. Piqueira is supported by FAPESP and CNPq. Luiz H. A. Monteiro is partially supported by CNPq