# Linear Approach for Synchronous State Stability  in Fully Connected PLL Networks

Authors: José R. C. Piqueira; Maurízio Q. de Oliveira; Luiz H. A. Monteiro
Journal: Mathematical Problems in Engineering
 (2008)

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## 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 withN nodes, the synchronous state is
locally asymptotically stable.


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## Body

## 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 connectedN-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].TheN 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 ωi and by its
phase ϕi(t). The whole network has its dynamics described by
phase errors and frequency errors defined by Δϕji(t) and Δϕ˙ji(t), respectively. Time delays τji 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 anN-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)
τN1: 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)
vN(t−τN1): VCO output of the node N with a delay τN1;(viii)
vd21(t): output of PD 2;(ix)
vd31(t): output of PD 3;(x)
vdN1(t): output of PD N;(xi)
vc1(t): output of F that controls the VCO.The instantaneous individual VCO phases and the
instantaneous phase errors,ϕi(t) and Δϕji(t), are expressed in the
forms
(2.1)ϕi(t)=ωit+θi(t),Δϕji(t)=ϕj(t)−ϕi(t).The node parameters related to PLL operation can be
characterized by the following:(i)
PD multiplying
factors:km1=km2=⋯=kmi=km, in volts−1, with i=1,2,…,N;(ii)
gains of the
VCOs:k1=k2=⋯=ki=k0, in rad/sV, with i=1,2,…,N;(iii)
cut-off
frequencies of F:μ11=μ12=μ1i=⋯=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 is(2.2)vi(t)=Vcosϕi(t);and signals received by the phase detector of node i from node j, with propagation delays τji, can be written as(2.3)vj(t−τji)=Vsin[ϕj(t−τji)],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]:(2.4)vdji(t)=kmvj(t−τji)vi(t),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 by(2.5)vdji(t)=kmV2[sinϕj(t−τji)cosϕi(t)].Each resulting signal given by
(2.5) is multiplied by 1/(N−1) and added, in
order to compose the filter input as below:(2.6)vdi(t)=(1N−1)(kmV22){∑j=1,j≠iNsin[ϕj(t−τji)−ϕi(t)]}.Defining kd=(1/2)(kmV2), expression (2.6) is simplified to(2.7)vdi(t)=kdN−1{∑j=1,j≠iNsin[ϕj(t−τji)−ϕi(t)]}.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 functionFi(s)=μ1/(s+μ1) [15], resulting in the
expression(2.8)v˙ci(t)+μ1vci(t)=μ1vdi(t).Replacingvdi, given by (2.7), and the VCO control signal vci=θ˙i(t)/k0 in (2.8), the
equation for the node phase is(2.9)θ¨i(t)+μ1θ˙i(t)−μ1(kokd(N−1)){∑j=1,j≠iNsin[ϕj(t−τji)−ϕi(t)]}=0.Then, considering ϕi(t)=wit+θi(t) in (2.9) and
defining μ2=kokd and k=μ1μ2/(N−1), the dynamics of each VCO phase in a fully connected
network is given by(2.10)ϕ¨i(t)+μ1ϕ˙i(t)−μ1ωi−k{∑j=1,j≠iNsin[ϕj(t−τji)−ϕi(t)]}=0.Equation (2.10) is similar to the
pendulum equation, containing a dissipation component μ1ϕ˙i(t), a delayed conservative term k∑j=1,j≠iNsin[ϕj(t−τji)−ϕi(t)], and a forcing part μ1ωi [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 ωs that can be
estimated. Thus, in order to estimate this frequency, the following hypotheses
are considered [13]:(a)
ϕ˙i(t)=ωs,(b)
ϕ¨i(t)=0,(c)
ϕj(t−τji)≈ϕj(t)+ϕ˙i(t)(−τji)≈ϕj(t)−ωsτji.Therefore,(2.11)μ1(ωs−ωi)−k[∑j=1,j≠iNsin(Δϕji−ωsτji)]=0.The values of (Δϕji−ωsτji) 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[Δϕji−ωsτji]≈Δϕji−ωsτji. Hence, for each node i,(2.12)μ1ωs−μ1ωi−k[∑j=1,j≠iNΔϕji]+kωs[∑j=1,j≠iNτji]=0.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 Δϕji is equal to
zero because Δϕji=−Δϕij, one can write(2.13)Nμ1ωs−μ1∑i=1Nωi+kωs(∑i=1N∑j=1,j≠iNτji)=0.Calculating ωs from
(2.13),(2.14)ωs=μ1(∑i=1Nωi)Nμ1+k(∑i=1N∑j=i,j≠iNτji).Dividing (2.14) by Nμ1, and replacing k=μ1μ2/(N−1), the estimation of the synchronous state frequency (ωs) is
obtained:(2.15)ωs=(1/N)(∑i=1Nωi)1+(μ2/N(N−1))(∑i=1N∑j=i,j≠iNτji).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, ωs is given by the
mean value of ωi. 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,Δϕji(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 nodes2,3,…,N and node 1, equations for Δϕj1(t), j=2,3,…,N are written as
below: (3.1)VCO 1:ϕ¨1(t)+μ1ϕ˙1(t)−μ1ω1−k×{sin[ϕ2(t−τ21)−ϕ1(t)]+sin[ϕ3(t−τ31)−ϕ1(t)]+sin[ϕ4(t−τ41)−ϕ1(t)]+⋯+sin[ϕN(t−τN1)−ϕ1(t)]}=0;VCO 2:ϕ¨2(t)+μ1ϕ˙2(t)−μ1ω2−k×{sin[ϕ1(t−τ12)−ϕ2(t)]+sin[ϕ3(t−τ32)−ϕ2(t)]+sin[ϕ4(t−τ42)−ϕ2(t)]+⋯+sin[ϕN(t−τN2)−ϕ2(t)]}=0;VCO 3:ϕ¨3(t)+μ1ϕ˙3(t)−μ1ω3−k×{sin[ϕ1(t−τ13)−ϕ3(t)]+sin[ϕ2(t−τ32)−ϕ3(t)]+sin[ϕ4(t−τ43)−ϕ3(t)]+⋯+sin[ϕN(t−τN3)−ϕ3(t)]}=0;⋮VCON−1:ϕ¨N−1(t)+μ1ϕ˙N−1(t)−μ1ωN−1−k×{sin[ϕ1(t−τ1N−1)−ϕN−1(t)]+sin[ϕ2(t−τ2N−1)−ϕN−1(t)]+sin[ϕ3(t−τ3N−1)−ϕN−1(t)]+⋯+sin[ϕN(t−τN−1N)−ϕN−1(t)]}=0;VCON:ϕ¨N(t)+μ1ϕ˙N(t)−μ1ωN−k×{sin[ϕ1(t−τ1N)−ϕN(t)]+sin[ϕ2(t−τ2N)−ϕN(t)]+sin[ϕ3(t−τ3N)−ϕN(t)]+⋯+sin[ϕN−1(t−τN−1N)−ϕN(t)]}=0.
Using the
condition ϕi(t−τji)≈ϕi(t)−ωsτji and considering Δϕj1(t)=ϕj(t)−ϕ1(t), then
(3.2)VCO 1:ϕ¨1(t)+μ1ϕ˙1−μ1ω1−k×{sin[Δϕ21(t)−ωsτ21]+sin[Δϕ31(t)−ωsτ31]+sin[Δϕ41(t)−ωsτ41]+⋯+sin[ΔϕN1(t)−ωsτN1]}=0;VCO 2:ϕ¨2(t)+μ1ϕ˙2(t)−μ1ω2−k×{sin[Δϕ12(t)−ωsτ12]+sin[Δϕ32(t)−ωsτ32]+sin[Δϕ42(t)−ωsτ42]+⋯+sin[ΔϕN2(t)−ωsτN2]}=0;VCO 3:ϕ¨3(t)+μ1ϕ˙3(t)−μ1ω3−k×{sin[Δϕ13(t)−ωsτ13]+sin[Δϕ23(t)−ωsτ23]+sin[Δϕ43(t)−ωsτ43]+⋯+sin[ΔϕN3(t)−ωsτN3]}=0;⋮VCON−1:ϕ¨N−1(t)+μ1ϕ˙N−1(t)−μ1ωN−1−k×{sin[Δϕ1N−1(t)−ωsτ1N−1]+sin[Δϕ2N−1(t)−ωsτ2N−1]+sin[Δϕ3N−1(t)−ωsτ3N−1]+⋯+sin[ΔϕNN−1(t)−ωsτNN−1]}=0;VCON:ϕ¨N(t)+μ1ϕ˙N(t)−μ1ωN−k×{sin[Δϕ1N(t)−ωsτ1N]+sin[Δϕ2N(t)−ωsτ2N]+sin[Δϕ3N(t)−ωsτ3N]+⋯+sin[ΔϕN−1N(t)−ωsτN−1N]}=0.
Expressing the
differences with node 1 as
reference,
(3.3)VCOs 1 and 2:ϕ¨2(t)−ϕ¨1(t)+μ1(ϕ˙2(t)−ϕ˙1(t))−μ1(ω2−ω1)−ksin[Δϕ12(t)−ωsτ12]−ksin[Δϕ32(t)+ωsτ32]−ksin[Δϕ42(t)−ω2τ42]−⋯−ksin[ΔϕN2(t)−ωsτN2]+ksin[Δϕ21(t)−ωsτ21]+⋯+ksin[ΔϕN1(t)−ωsτN1]=0;VCOs 1 and 3:ϕ¨3(t)−ϕ¨1(t)+μ1(ϕ˙3(t)−ϕ˙1(t))−μ1(ω3−ω1)−ksin[Δϕ13(t)−ωsτ13]−ksin[Δϕ23(t)−ωsτ23]−ksin[Δϕ43(t)−ω2τ43]−⋯−ksin[ΔϕN3(t)−ωsτN3]+ksin[Δϕ21(t)+ωsτ21]+ksin[Δϕ31(t)+ωsτ31]+ksin[Δϕ41(t)+ωsτ41]+⋯+ksin[ϕN1(t)−ωsτN1]=0;⋮VCOs 1 andN:ϕ¨N(t)−ϕ¨1(t)+μ1(ϕ˙N(t)−ϕ˙1(t))−μ1(ωN−ω1)−ksin[Δϕ1N(t)−ωsτ1N]−ksin[Δϕ2N(t)−ωsτ2N]−ksin[Δϕ3N(t)−ω2τ3N]−⋯−ksin[ΔϕN−1N(t)−ωsτN−1N]+ksin[Δϕ21(t)+ωsτ21]+ksin[Δϕ31(t)+ωsτ31]+ksin[Δϕ41(t)+ωsτ52]+⋯+ksin[ϕN1(t)−ωsτN1]=0.
Replacing the
terms Δϕj1(t)=ϕj(t)−ϕ1(t), Δϕ˙j1(t)=ϕ˙j(t)−ϕ˙1(t), Δϕ¨j1(t)=ϕ¨j(t)−ϕ¨1(t), and using the identity sin[Δϕj1(t)−ωsτj1]+sin[Δϕji(t)+ωsτji]=2sinΔϕjicosωsτji in the former
expressions, the VCO phase differences become
(3.4)VCOs 1 and 2:Δϕ¨21(t)+μ1Δϕ˙21(t)−μ1Δω21+2ksinΔϕ21(t)cos(ωsτ21)−ksin[Δϕ32(t)+ωsτ32]−ksin[Δϕ42(t)−ω2τ42]−⋯−ksin[ΔϕN2(t)−ωsτN2]+ksin[Δϕ31(t)−ωsτ31]+ksin[Δϕ41(t)−ωsτ41]+⋯+ksin[ΔϕN1(t)−ωsτN1]=0;VCOs 1 and 3:Δϕ¨31(t)+μ1Δϕ˙31(t)−μ1Δω31+2ksinΔϕ31(t)cos(ωsτ31)−ksin[Δϕ23(t)+ωsτ23]−ksin[Δϕ43(t)−ω2τ43]−⋯−ksin[ΔϕN3(t)−ωsτN3]+ksin[Δϕ21(t)−ωsτ21]+ksin[Δϕ31(t)−ωsτ31]+⋯+ksin[ΔϕN1(t)−ωsτN1]=0;⋮VCOs 1 andN:Δϕ¨N1(t)+μ1Δϕ˙N1(t)−μ1ΔωN1+2ksinΔϕN1(t)cos(ωsτN1)−ksin[Δϕ2N(t)+ωsτ2N]−ksin[Δϕ3N(t)−ω2τ3N]−⋯−ksin[ΔϕN−1N(t)−ωsτN−1N]+ksin[Δϕ21(t)−ωsτ21]+ksin[Δϕ31(t)−ωsτ31]+⋯+ksin[ΔϕN−11(t)−ωsτN−11]=0.
These equations
give the general expression that describes the dynamical behavior of the phase
differences for the network VCOs as below:(3.5)Δϕ¨j1(t)+μ1Δϕ˙j1(t)−μ1Δωj1+2ksinΔϕj1(t)cos(ωsτj1)−k{∑m=2,m≠jNsin[Δϕmj(t)−ωsτmj]}+k{∑m=2,m≠jNsin[Δϕm1(t)−ωsτm1]}=0.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)(3.6)Δϕ¨j1(t)+μ1Δϕ˙j1(t)−μ1Δωj1+2kΔϕj1(t)−k{∑m=2,m≠jN[Δϕmj(t)−Δϕm1(t)]}+kωs{∑m=1,m≠jN[τmj−τm1]}=0.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 Section2,
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 areΔϕ12 and Δϕ13. By using (3.5), the system is described by
(4.1)VCOs 1 and 2:Δϕ¨12+μ1Δϕ˙12+2kΔϕ12+kΔϕ13−kΔϕ23+μ1Δω21+kωs(τ13−τ23)︸a=0;VCOs 1 and 3:Δϕ¨13+μ1Δϕ˙13+2kΔϕ13+kΔϕ12+kΔϕ23+μ1Δω31+kωs(τ12−τ32)︸b=0.
The system
composed of two second-order equations (4.1) can be described by the
state variables:(4.2)x1≡Δϕ12,x3≡Δϕ13,x2≡Δϕ˙12,x4≡Δϕ˙13.Considering that Δϕ23=x3−x1, the following first-order equations
result:(4.3)x˙1=x2,x˙2=−3kx1−μ1x2−a,x˙3=x4,x˙4=−3kx3−μ1x4−b.Consequently, the Jacobian matrixJ from (4.3)
is(4.4)J=[0100−3k−μ100000100−3k−μ1].The eigenvalues of J are the roots
of the characteristic polynomial
(4.5)(λ2+μ1λ+3k)2=0,which have
multiplicity two, and are given by(4.6)λ1=−μ12+μ12−12k2,λ2=−μ12−μ12−12k2.


## 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 byΔϕ12, Δϕ13, and Δϕ14. By using (3.5), the system is written as
(4.7)VCOs 1 and 2:Δϕ¨12+μ1Δϕ˙12+2kΔϕ12+kΔϕ13−kΔϕ23+kΔϕ14−kΔϕ24+μ1Δω21+kωs(τ13−τ23+τ14−τ24)︸a=0;VCOs 1 and 3:Δϕ¨13+μ1Δϕ˙13+2kΔϕ13+kΔϕ12+kΔϕ23+kΔϕ14−kΔϕ34+μ1Δω31+kωs(τ12−τ32+τ14−τ34)︸b=0;VCOs 1 and 4:Δϕ¨14+μ1Δϕ˙14+2kΔϕ14+kΔϕ12+kΔϕ24+kΔϕ13+kΔϕ34+μ1Δω41+kωs(τ12−τ42+τ13−τ43)︸c=0.
The
second-order expressions (4.7) can be rewritten in terms of state
variables:(4.8)x1≡Δϕ12,x3≡Δϕ13,x5≡Δϕ14,x2≡Δϕ˙12,x4≡Δϕ˙13,x6≡Δϕ˙14.Considering that Δϕ23=x3−x1, Δϕ24=x5−x1, and Δϕ34=x3−x5, the following first-order equations
result:(4.9)x˙1=x2,x˙2=−4kx1−μ1x2−a,x˙3=x4,x˙4=−4kx3−μ1x4−b,x˙5=x6,x˙6=−4kx5−μ1x6−c.Consequently,
the Jacobian matrixJ from (4.9)
is(4.10)J=[010000−4k−μ1000000010000−4k−μ1000000010000−4k−μ1].The eigenvalues of J are given by
the roots of the characteristic polynomial:
(4.11)(λ2+μ1λ+4k)3=0,which have
multiplicity 3, and are(4.12)λ1=−μ12+μ12−16k2,λ2=−μ12−μ12−16k2.


## 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 anN-node network.
nEigenvalues and multiplicitiesPolynomialλ1mλ2m2(λ2+μ1λ+2k)−μ12+μ12−8k21−μ12−μ12−8k213(λ2+μ1λ+3k)2−μ12+μ12−12k22−μ12−μ12−12k224(λ2+μ1λ+4k)3−μ12+μ12−16k23−μ12−μ12−16k235(λ2+μ1λ+5k)4−μ12+μ12−20k24−μ12−μ12−20k246(λ2+μ1λ+6k)5−μ12+μ12−246k25−μ12−μ12−24k257(λ2+μ1λ+7k)6−μ12+μ12−28k26−μ12−μ12−28k26⋮⋮⋮⋮⋮⋮N(λ2+μ1λ+Nk)(N−1)−μ12+μ12−4Nk2(N−1)−μ12−μ12−4Nk2(N−1)It can be observed that for any number of nodes, the
eigenvalues from Table1 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 Table1 can be
used for determining the transient behavior of such a resynchronization
process.



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