Fully Connected PLL Networks: How Filter Determines the Number of Nodes

Abstract

Synchronization plays an important role in telecommunication systems, integrated circuits, and automation systems. Formerly, the masterslave synchronization strategy was used in the great majority of cases due to its reliability and simplicity. Recently, with the wireless networks development, and with the increase of the operation frequency of integrated circuits, the decentralized clock distribution strategies are gaining importance. Consequently, fully connected clock distribution systems with nodes composed of phase-locked loops (PLLs) appear as a convenient engineering solution. In this work, the stability of the synchronous state of these networks is studied in two relevant situations: when the node filters are first-order lag-lead low-pass or when the node filters are second-order low-pass. For first- order filters, the synchronous state of the network shows to be stable for any number of nodes. For second-order filter, there is a superior limit for the number of nodes, depending on the PLL parameters.

1. Introduction

Synchronization networks assure the correct temporal order of information processing in communication systems, computation and control and many commercial systems adopt a masterslave synchronization strategy due to its reliable behavior, construction facility, and low cost [1–3]. In this architecture, a master node dictates the phase and frequency scales for the entire network.

On the other hand, in a fully connected architecture, phase and frequency scales are determined by all the nodes as they are all connected to each other. The main advantage of the fully connected architecture is its robustness when nodes are added or dropped [4]. Nowadays, besides being used in telecommunication networks, the fully connected architecture is used in digital electronic circuits for clock signal distribution [5–7], and in synchronous neural networks for pattern recognition [8–10].

Modeling fully connected networks result in nonlinear coupled oscillations. Some interesting works about this kind of architecture were developed between 1960 and 1980 [11–15]. These papers were written when the service integration started to be taken into account for telecommunication networks and present several robust results for clock distribution systems, mainly considering linear analogous phase detection and linear coupling between nodes.

In these works, synchronous state frequencies and their stability conditions for fully connected PLL networks were presented depending on the free-running frequencies and node phase detector gains.

Trying to extend the results to the case with nonlinear coupling, in [16, 17], expressions for the synchronous state and conditions for its stability are derived, considering analog phase-detectors and first-order lag low-pass filters. The synchronous state of a fully connected network showed to be locally asymptotically stable for any number 𝑁 of nodes.

As, sometimes, double-frequency jitter degrades the performance of the PLLs [18–20], filters are designed more accurately [21, 22].

Here, a nonlinear state space model of fully connected PLL networks with 𝑁 nodes, considering a first-order lead-lag low-pass and a second-order low-pass filters in the nodes is developed, trying to minimize double-frequency distortion [23, 24].

First, models for the fully connected PLL network with first-order lag-lead node filters and second-order low-pass node filters are developed. Then, by using Taylor series development around the synchronous state [25], it was shown that for the lag-lead node filters, the synchronous state is locally asymptotically stable for any number of nodes, in the same way of the lag node filters. For third-order node filters, a local Hopf bifurcation that appears in the single node dynamics [26–29] gives a superior limit for the number of nodes in the network.

The contribution is about analytic modeling of this kind of network and experimental and numerical works can be conducted trying to confirm the results as suggested in [27, 30–32].

2. Fully Connected PLL Network Model

In a fully connected PLL network with 𝑁 nodes, each node has 𝑁−1 inputs corresponding to the phase comparisons, as shown in Figure 1, with the nodes mutually coupled by using two-way connections [30, 31]. As there is no integrated circuit with this architecture, the implementation of the nodes requires a combination of 𝑁−1 PLL chips with their phase detector outputs weighted and being the input of the filter 𝑓^(𝑗)(𝑡) that feeds the local VCO [17, 30–32].

Figure 1 

PLL block diagram in a fully connected architecture.

For a given node 𝑗, the output 𝑣_𝑜^(𝑗)(𝑡), 𝑗=1,2,…,𝑁, and input signals 𝑣_𝑜^(ℓ)(𝑡), ℓ=1,…,𝑗−1,𝑗+1,…,𝑁, are given by 𝑣_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)=v_𝑜^(ℓ)sin(𝜔_𝑀(𝑡−𝜏_ℓ,𝑗)+𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗𝑣)),_𝑜(𝑗)(𝑡)=v_𝑜^(𝑗)cos(𝜔_𝑀𝑡+𝜃_𝑜^(𝑗)(𝑡)) (2.1)(2.2) where v_𝑜^(ℓ) and v_𝑜^(𝑗) are the amplitudes of the input and output signals, respectively; 𝜔_𝑀 is the free-running angular frequency of all VCOs, 𝜃_𝑜^(𝑗)(𝑡) is the instantaneous phase of node 𝑗, 𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗) is the phase of node ℓ delayed by the signal propagation time from node ℓ to 𝑗, 𝜏_ℓ,𝑗.

Assuming that all VCOs in the network have the same free-running angular frequency, the phase and frequency spatial errors can be defined as 𝜗^(ℓ,𝑗)(𝑡)=𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)−𝜃_𝑜^(𝑗)𝜗̇(𝑡),_(ℓ,𝑗)(𝑡)=𝜃̇_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)−𝜃̇_𝑜^(𝑗)(𝑡). (2.3)(2.4) Considering the expressions of the phase errors between node 𝑗 and of all other nodes as given by (2.3) and summing them, the estimation of phase 𝜃_𝑜^(𝑗)(𝑡), of the node 𝑗, is written as follows: 𝜃_𝑜^(𝑗)1(𝑡)=𝑁−1^𝑁_{ℓ=1ℓ≠𝑗}(𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)−𝜗^(ℓ,𝑗)(𝑡)). (2.5) The PLLs are described by a differential equation of order 𝑃^(𝑗)+1, considering that the order of the linear filter 𝑓(𝑡) is 𝑃^(𝑗) [21, 22]. The transfer function 𝐹^(𝑗)(𝑠) of the node 𝑗 filter can be expressed in the form of a rational fraction of polynomials: 𝐹^(𝑗)𝑁(𝑠)=^(𝑗)(𝑠)𝐷^(𝑗)(𝑠), (2.6) with 𝑁^(𝑗)(𝑠)=^𝑀(𝑗)_𝑚=0𝛼_𝑚^(𝑗)𝑠^𝑚𝐷_(𝑗),(𝑠)=^𝑃(𝑗)_𝑝=0𝛽_𝑝^(𝑗)𝑠^𝑝, (2.7)(2.8) considering that 𝑀^(𝑗)≤𝑃^(𝑗).

A consequence from the fact that the filter can be modeled by (2.6) and from the structure of (2.1) and (2.2) is that the network dynamics can be expressed in a simpler way considering the operators 𝑄^(𝑗)[⋅] and 𝐿^(𝑗)[⋅], and the node gains 𝐺^(ℓ,𝑗) as follows: 𝑄^(𝑗)[⋅]=^𝑀(𝑗)_𝑚=0𝛼_𝑚^(𝑗)𝑑^𝑚𝑑𝑡^𝑚𝐿(⋅),_(𝑗)[⋅]=^𝑃(𝑗)_𝑝=0𝛽_𝑝^(𝑗)𝑑^𝑝+1𝑑𝑡^𝑝+1𝐺(⋅),_(ℓ,𝑗)=12𝑎_ℓ,𝑗𝑘_𝑚^(𝑗)𝑘_𝑜^(𝑗)v_𝑜^(𝑗)v_𝑜^(ℓ). (2.9) In these definitions, the positive numbers 𝑘_𝑚^(𝑗) are the phase detector gains, 𝑘_𝑜^(𝑗), the VCO control gains and 𝑎_ℓ,𝑗, the weighting factors between nodes ℓ and 𝑗.

Therefore, the filter inputs are 𝑣_𝑑^(𝑗)1(𝑡)=𝑁−1^𝑁_{ℓ=1ℓ≠𝑗}𝑎_ℓ,𝑗𝑢_𝑑^(ℓ,𝑗)(𝑡), (2.10) with 1𝑁−1^𝑁_{ℓ=1ℓ≠𝑗}𝑎_ℓ,𝑗=1. (2.11) The outputs of the phase detectors are proportional to the product between their input and output signals: 𝑢_𝑑^(ℓ,𝑗)(𝑡)=𝑘_𝑚^(𝑗)𝑣_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)𝑣_𝑜^(𝑗)(𝑡), (2.12) resulting 𝑢_𝑑^(ℓ,𝑗)𝐺(𝑡)=^(ℓ,𝑗)𝑎_ℓ,𝑗𝑘_𝑜^(𝑗)𝑢_𝛿^(ℓ,𝑗)(𝑡), (2.13) with 𝑢_𝛿^(ℓ,𝑗)(𝑡)=sin(𝜗^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗)+sin(2𝜔_𝑀𝑡+𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗)+𝜃_𝑜^(𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗). (2.14) Solving (2.3) for 𝜃_𝑜^(𝑗)(𝑡), replacing the result in (2.14), and doing some algebra: 𝑢_𝛿^(ℓ,𝑗)(𝑡)=sin(𝜗^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗)+sin(2(𝜔_𝑀𝑡+𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗))−𝜗^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗). (2.15) The output phase 𝜃_𝑜^(𝑗)(𝑡) of the VCO is controlled according to 𝑑𝜃𝑑𝑡^𝑜(𝑗)(𝑡)=𝑘_𝑜^(𝑗)𝑣_𝑐^(𝑗)(𝑡), (2.16) with control signal 𝑣_𝑐^(𝑗) being the filter response to the weighted error signal 𝑣_𝑑^(𝑗), and given by 𝑣_𝑐^(𝑗)(𝑡)=𝑓^(𝑗)(𝑡)∗𝑣_𝑑^(𝑗)(𝑡). (2.17) Consequently, replacing (2.17) in (2.16) and applying the convolution theorem [33]: 𝑘_𝑜^(𝑗)𝑁^(𝑗)(𝑠)𝑉_𝑑^(𝑗)(𝑠)=𝑠𝐷^(𝑗)(𝑠)Θ_𝑜^(𝑗)(𝑠). (2.18) Considering the operators 𝑄^(𝑗)[⋅] and 𝐿^(𝑗)[⋅] and using (2.5), (2.10), and (2.13), 𝐿^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝜗^(ℓ,𝑗)(𝑡)+𝑄^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝐺^(ℓ,𝑗)𝑢_𝛿^(ℓ,𝑗)(𝑡)=𝐿^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗). (2.19) Replacing (2.15) in (2.19) 𝐿^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝜗^(ℓ,𝑗)(𝑡)+𝑄^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝐺^(ℓ,𝑗)sin(𝜗^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗)=−𝑄^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝐺^(ℓ,𝑗)sin(2(𝜔_𝑀𝑡+𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗))−𝜗_𝑜^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗)+𝐿^𝑁(𝑗)_{ℓ=1ℓ≠𝑗}𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗), (2.20) for 𝑗=1,2,…,𝑁.

Finally, (2.20) models the fully connected PLL network, neglecting the second-harmonic term that is supposed to be filtered [3, 23, 24]. To simplify the analytical reasoning, for a given node 𝑗, it is considered that all the phase errors are equally weighted, and that all the constitutive parameters of the nodes are the same. These assumptions do not change the main conclusions as weight criterion and node parameters affect only the system transient behavior. Therefore, from now on 𝐺^(ℓ,𝑗)=𝐺, 𝐿^(𝑗)[⋅]=𝐿[⋅], 𝑄^(𝑗)[⋅]=𝑄[⋅] and 𝐿^𝑁_{ℓ=1ℓ≠𝑗}𝜗^(ℓ,𝑗)(𝑡)+𝐺𝑄^𝑁_{ℓ=1ℓ≠𝑗}sin(𝜗^(ℓ,𝑗)(𝑡)−𝜔_𝑀𝜏_ℓ,𝑗)=𝐿^𝑁_{ℓ=1ℓ≠𝑗}𝜃_𝑜^(ℓ)(𝑡−𝜏_ℓ,𝑗). (2.21) The fixed transmission delays, 𝜏_ℓ,𝑗, are also neglected, since they only shift the equilibrium points from its original position in equation not affecting their stability conditions (2.21). Using (2.5) −(𝑁−1)𝐿𝜃_𝑜^(𝑗)(𝑡)+𝐺𝑄^𝑁_{ℓ=1ℓ≠𝑗}sin(𝜗^(ℓ,𝑗)(𝑡))=0, (2.22) for 𝑗=1,2,…,𝑁.

From (2.5) it can be seen that, in the synchronous state the phase of node 𝑗 is linearly dependent on the phase output of all the other nodes. Then, it is necessary to obtain a set of 𝑁−1 linearly independent state variables [31].

This is obtained by rewriting (2.22) for node 1 and for the other nodes as follows: −(𝑁−1)𝐿𝜃_𝑜⁽¹⁾(𝑡)+𝐺𝑄^𝑁_ℓ=2sin(𝜗^(ℓ,1)(𝑡))=0, (2.23) for node 1, and −(𝑁−1)𝐿𝜃_𝑜^(𝑗)(𝑡)+𝐺𝑄^𝑁_{ℓ=1ℓ≠𝑗}sin(𝜗^(ℓ,𝑗)(𝑡))=0, (2.24) for 𝑗=2,…,𝑁.

Expressing the phase difference between all the other nodes and the node 1, that is, subtracting (2.24) from (2.23), and considering (2.3) with 𝜏_ℓ,𝑗=0(𝑁−1)𝐿𝜗^(𝑗,1)(𝑡)+𝐺𝑄^𝑁_ℓ=2sin(𝜗^(ℓ,1)(𝑡))−^𝑁_{ℓ=1ℓ≠𝑗}sin(𝜗^(ℓ,𝑗)(𝑡))=0. (2.25) Considering that 𝜗^(1,𝑗)(𝑡)=𝜃_𝑜⁽¹⁾(𝑡)−𝜃_𝑜^(𝑗)(𝑡)=−(𝜃_𝑜^(𝑗)(𝑡)−𝜃_𝑜⁽¹⁾(𝑡))=−𝜗^(𝑗,1)(𝑡) (2.26) and that 𝜗^(ℓ,𝑗)(𝑡)=𝜃_𝑜^(ℓ)(𝑡)−𝜃_𝑜^(𝑗)(𝑡)=𝜃_𝑜^(ℓ)(𝑡)−𝜃_𝑜⁽¹⁾(𝑡)+𝜃_𝑜⁽¹⁾(𝑡)−𝜃_𝑜^(𝑗)(𝑡)=𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡), (2.27) working on (2.25), 𝐿𝜗^(𝑗,1)(𝑡)+2𝜇₁𝑄sin(𝜗^(𝑗,1)(𝑡))+𝜇₁𝑄^𝑁_{ℓ=2ℓ≠𝑗}(sin(𝜗^(ℓ,1)(𝑡))−sin(𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡)))=0, (2.28) for 𝑗=2,…,𝑁, and 𝜇₁=𝐺𝑁−1. (2.29) The set of 𝑁−1 equations given by (2.28) is the general model that describes the dynamical behavior of the phase differences for the network nodes. The use of the lag filter, as in [16, 17], simplifies the mathematical reasoning. However, this kind of filter is useful only when the bandwidth, that is, the difference between input signal frequencies and central operation frequency of the PLL does not need to be narrow [34].

In practical cases, when narrow bandwidth and large gain are simultaneously required, lead-lag filters with transfer functions given by 𝐹(𝑠)=𝑠+𝛼₀𝑠+𝛽₀ (2.30) are used [34].

The second-order filter, with transfer function given by 𝐹(𝑠)=𝑠+𝛼₀𝑠²+𝛽₁𝑠+𝛽₀, (2.31) prevents spurious side-bands by reducing the influence of double-frequency jitter and transmission noise [21–24, 34], in spite of provoking bifurcation [26, 27].

Considering a first-order lead-lag filter in each node of the PLL network and (2.28),

𝜗̈^(𝑗,1)(𝑡)+𝛽₀𝜗̇^(𝑗,1)(𝑡)+2𝜇₁𝜗̇^(𝑗,1)(𝑡)cos(𝜗^(𝑗,1)(𝑡))+2𝛼₀𝜇₁sin(𝜗^(𝑗,1)(𝑡))+𝜇₁^𝑁_{ℓ=2ℓ≠𝑗}(𝜗̇^(ℓ,1)(𝑡)cos(𝜗^(ℓ,1)(𝑡))−(𝜗̇^(ℓ,1)(𝑡)−𝜗̇^(𝑗,1)(𝑡))cos(𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡)))+𝛼₀𝜇₁^𝑁_{ℓ=2ℓ≠𝑗}(sin(𝜗^(ℓ,1)(𝑡))−sin(𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡)))=0. (2.32) for 𝑗=2,…,𝑁.

For the second-order filter, the model of the network is given by 𝜗⃛^(𝑗,1)(𝑡)+𝛽₁𝜗̈^(𝑗,1)(𝑡)+𝛽₀𝜗̇^(𝑗,1)(𝑡)+2𝜇₁𝜗̇^(𝑗,1)(𝑡)cos(𝜗^(𝑗,1)(𝑡))+2𝛼₀𝜇₁sin(𝜗^(𝑗,1)(𝑡))+𝜇₁^𝑁_{ℓ=2ℓ≠𝑗}(𝜗̇^(ℓ,1)(𝑡)cos(𝜗^(ℓ,1)(𝑡))−(𝜗̇^(ℓ,1)(𝑡)−𝜗̇^(𝑗,1)(𝑡))cos(𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡)))+𝛼₀𝜇₁^𝑁_{ℓ=2ℓ≠𝑗}(sin(𝜗^(ℓ,1)(𝑡))−sin(𝜗^(ℓ,1)(𝑡)−𝜗^(𝑗,1)(𝑡)))=0, (2.33) for 𝑗=2,…,𝑁.

Therefore, (2.32) models the fully connected network considering first-order lead-lag node filters. Similarly, (2.33) represents the network with second-order node filters.In order to simplify the study of the equilibrium and its stability for (2.32) and (2.33),

the next section presents state space models for both cases.

3. State Space Equations

From (2.32), state equations can be established considering the phase and frequency errors, measured taking node 1 as reference, that is, 𝑥₁^(𝑚)=𝜗^(𝑗,1)𝑥(𝑡),_2(𝑚)=𝜗̇^(𝑗,1)(𝑡), (3.1) for 𝑗=2,…,𝑁 and 𝑚=𝑗−1.

Then, considering a first-order lag-lead low-pass filter, state equations are 𝑥̇₁^(𝑚)=𝑥₂^(𝑚)𝑥̇₂^(𝑚)=−𝛽₀𝑥₂^(𝑚)−2𝜇₁𝑥₂^(𝑚)cos(𝑥₁^(𝑚))−2𝛼₀𝜇₁sin(𝑥₁^(𝑚))−𝜇₁^𝑁−1_{𝑛=1𝑛≠𝑚}(𝑥₂^(𝑛)cos(𝑥₁^(𝑛))−(𝑥₂^(𝑛)−𝑥₂^(𝑚))cos(𝑥₁^(𝑛)−𝑥₁^(𝑚)))−𝛼₀𝜇₁^𝑁−1_{𝑛=1𝑛≠𝑚}(sin(𝑥₁^(𝑛))−sin(𝑥₁^(𝑛)−𝑥₁^(𝑚))), (3.2) for 𝑚=1,2,…,𝑁−1.

Defining a new state variable, 𝑥₃^(𝑚)=𝜗̈^(𝑗,1)(𝑡), (3.3) and by using (2.33), state equations considering a second-order low-pass filter are derived as 𝑥̇₁^(𝑚)=𝑥₂^(𝑚)𝑥̇₂^(𝑚)=𝑥₃^(𝑚)𝑥̇₃^(𝑚)=−𝛽₁𝑥₃^(𝑚)−𝛽₀𝑥₂^(𝑚)−2𝜇₁𝑥₂^(𝑚)cos(𝑥₁^(𝑚))−2𝛼₀𝜇₁sin(𝑥₁^(𝑚))−𝜇₁^𝑁−1_{𝑛=1𝑛≠𝑚}(𝑥₂^(𝑛)cos(𝑥₁^(𝑛))−(𝑥₂^(𝑛)−𝑥₂^(𝑚))cos(𝑥₁^(𝑛)−𝑥₁^(𝑚)))−𝛼₀𝜇₁^𝑁−1_{𝑛=1𝑛≠𝑚}(sin(𝑥₁^(𝑛))−sin(𝑥₁^(𝑛)−𝑥₁^(𝑚))), (3.4) for 𝑚=1,2,…,𝑁−1.

Expressions (3.2) and (3.4) describe the dynamics of the phase adjustments of a fully- connected network depending on the node parameters 𝜇₁, 𝛼₀, 𝛽₀ and 𝛽₁, considering first-order lag-lead and second-order filters, respectively. These equations allow the research of the synchronous state stability conducted in the next section.

4. Synchronous State Stability

The fully connected PLL network reaches the synchronous state when all the frequency errors vanish. Furthermore, the robustness of this state under small perturbations gives an idea about the network steady state operation.

In this section, the local stability of the synchronous state, for the two different filter types, by using the Taylor series representing the vector fields given by (3.2) and (3.4) in a neighborhood of the synchronous state is studied.

Calculating the eigenvalues of the Jacobian of the vector fields that represents their linear part, conditions for local asymptotic stability of the synchronous state are derived [25].

4.1. First-Order Lag-Lead Filter Analysis

The equilibrium points of the state equations in (3.2) that correspond to the synchronous states of the network are 𝐱^∗(𝑚)=𝑘𝜋0^𝑇, (4.1) where 𝑘=0,±1,±2,… ; and for 𝑚=1,2,…,𝑁−1, then 𝐱^∗=𝐱^∗(1)𝑇𝐱^∗(2)𝑇⋯𝐱^{∗(𝑁−1)𝑇𝑇}. (4.2) For odd values of 𝑘, the equilibrium points are unstable for any parameter combination [17]. The stability of the equilibrium points corresponding to even values of 𝑘 can be studied by the eigenvalues of (4.3), considering the equilibrium point 𝐱^∗, with 𝑘=0: 𝐀=01⋯00−𝛼₀𝑁𝜇₁−𝛽₀−𝑁𝜇₁⋯00⋮⋮⋱⋮⋮00⋯0100⋯−𝛼₀𝑁𝜇₁−𝛽₀−𝑁𝜇₁. (4.3) The characteristic polynomial of the matrix 𝐀 is 𝑃(𝜆)=(𝜆²+(𝛽₀+𝑁𝜇₁)𝜆+𝛼₀𝑁𝜇₁)^𝑁−1, (4.4) that presents 𝑁−1 repeated pair of roots given by 𝛽𝜆=−⁰+𝑁𝜇₁2±(𝛽₀+𝑁𝜇₁)²−4𝛼₀𝑁𝜇₁2. (4.5) As all physically possible parameters in (4.5) are positive real numbers, these roots are either real negative or complex conjugated with negative real part. Consequently, for any number of nodes and any parameter combination, the fully connected PLL network with first-order lag-lead filters presents locally asymptotically stable synchronous states.

4.2. Second-Order Filter Analysis

The equilibrium points of the state equations in (3.4) are 𝐱^∗(𝑚)=𝑘𝜋00^𝑇, (4.6) where 𝑘=0,±1,±2,… ; and for 𝑚=1,2,…,𝑁−1, then 𝐱^∗=𝐱^∗(1)𝑇𝐱^∗(2)𝑇⋯𝐱^{∗(𝑁−1)𝑇𝑇}. (4.7) For odd values of 𝑘, the equilibrium points are unstable for any parameter combination [27]. The stability of the equilibrium points corresponding to even values of 𝑘 can be studied by the eigenvalues of (4.8), considering the equilibrium point 𝐱^∗, with 𝑘=0: 𝐀=010⋯000001⋯000−𝛼₀𝑁𝜇₁−𝛽₀−𝑁𝜇₁−𝛽₁⋯000⋮⋮⋮⋮⋱⋮⋮000⋯010000⋯001000⋯−𝛼₀𝑁𝜇₁−𝛽₀−𝑁𝜇₁−𝛽₁. (4.8) The characteristic polynomial of the matrix 𝐀 is 𝑃(𝜆)=(𝜆³+𝛽₁𝜆²+𝛽₀+𝑁𝜇₁)𝜆+𝛼₀𝑁𝜇₁)^𝑁−1. (4.9) In order to verify if (4.9) presents roots with positive real part, Routh-Hurwitz criterion ([35]) is applied as follows:

𝜆³1𝛽₀+𝑁𝜇₁𝜆²𝛽₁𝛼₀𝑁𝜇₁𝜆¹𝛽₁(𝛽₀+𝑁𝜇₁)−𝛼₀𝑁𝜇₁𝛽₁𝜆⁰𝛼₀𝑁𝜇₁. (4.10) As all physically possible parameters are positive real numbers, the roots of (4.9) are either real negative or complex conjugated with negative real part only if 𝛽𝑁<¹𝛽₀𝜇₁(𝛼₀−𝛽₁). (4.11) Consequently, (4.11) gives a superior limit for the number of nodes in a fully connected PLL network with second-order node filters. Above this value, all the synchronous states of the network are unstable

5. Conclusions

fully connected PLL networks can have their performance improved by choosing filters different from first-order lag low-pass. If first-order lag-lead low-pass are chosen, the synchronous states of the network are locally asymptotically stable for any number of nodes. The adjustment of the parameters 𝛼₀ and 𝛽₀ only changes the transient responses, allowing the desired bandwidth.

If a second-order low pass is used, in order to further improve the transient response, (4.11) establishes a superior limitation for the number of nodes of the network.