critical delays in clock distribution circuits

7/27/2019 Critical Delays in Clock Distribution Circuits

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Hindawi Publishing CorporationMathematical Problems in EngineeringVolume , Article ID ,pageshttp://dx.doi.org/.//

Research ArticleCritical Delays in Clock Distribution Circuits

Jos R. C. Piqueira and Diego Paolo F. Correa

Escola Politecnica da Universidade de Sao Paulo, Avenida Professor Luciano Gualberto, Travessa ,No. , - Sao Paulo, SP, Brazil

Correspondence should be addressed to Jose R. C. Piqueira; [email protected]

Received August ; Revised September ; Accepted October

Academic Editor: Jui-Sheng Lin

Copyright J. R. C. Piqueira and D. P. F. Correa. Tis is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

As the distribution o clock signals between the nodes o a network became a critical operational requisite, the transmission delayshave to be studied because they affect the accurate recovering o the time basis. In this work, the critical delay value is calculatedconsidering simplications compatible with practical situations. Te main contribution is to consider the dissipative terms in thenode equations, expressing critical delays, depending on the time constant o the dissipation.

1. Introduction

Te phase control problem in a network o oscillators is aproblem that appears in several engineering situations: worldwide time signal distribution, synchronous communicationnetworks, distributed computation, processes control, andsynchronous operation o micro- and nanocircuits. Generallyspeaking, the distribution o clock signals can be perormedcontrolling the phase o a node oscillation by using theinormation o the phases o the other nodes that composethe whole network.

Modeling this kind o network presentstwo types o clockdistribution strategies: master-slave (MS) and ull-connected(FC), described by Lindsey et al., in a seminal paper that isan important reerence or engineers working on design o

networks [].Concerning the networks conceived up to the end o

the last century, Synchronous Digital Hierarchy (SDH),employing MS strategies, was the most important solutionbecoming a standard recommended by IU (Internationalelecommunication Union) []. Te mathematical analysiso this solution has been widely studied with a lot o differentapproaches: linear differential equations [], nonlinear differ-ential equations [], and stochastic differential equations [].

Nowadays, it could be considered that clock signal dis-tribution by using MS strategies is well studied and networkdesigners have a loto goodreerences availableor the severaltypes o phase detection and signal ltering [].

However, the network architecture evolved to complex

topologies with phase and requency reerences depending,at each node, on the phase and requency o almost all theother nodes [] and, consequently, to obtain general solutionsbecomes easible only or certain particular situations [] asthe resulting equations or the isolated nodes are second-order, including a dissipative term. Besides, the couplingterms between the nodes are composed o nonlinear delayedterms [].

Here, considering that operational requencies o thenode oscillators are normalized, theclock signalphase o eachnode is supposed to be described by []

()+1

()=

sin

()

() +

() . ()

In (), represents a dissipative time constant term, andrepresents a requency node gain, both related to node. Teeffect o weak orces is represented by() that considersnoise and small perturbations. A wave propagation term

appears as() and it is responsible or considering thedependence o the local phase on the phase o all nodes andis given by

()= =1

, , . ()


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Mathematical Problems in Engineering

1

1.5

2

2.5

T

w1,2

w1

w2

3 2/4 5/4 3/2 7/4

F : Real positive values or()given in (). Tese valuesappear or > 3/4.In (),,represents the normalized connection matrix, thatis,=1(,) = 1, or each node, and, is the propagationdelay between the nodesand.

Te main goal o this paper is to study the severalpossible behaviors o a network, trying to express them tobe related to the connection matrix, the nodes ree-runningrequencies, and the delays between the nodes. Consideringthe complexityo the problem, some simpliying assumptionsare necessary in order to treat the problem.

Te rst simplication is about the network structurethatis supposed to be triangular [] as the practical networksare trees, composed o cells that can be modeled as triangles

[,]. Tey are supposed to have the same parameters. Inspite o not being a general case, this assumption providesa simple reasoning with useul practical conclusions, as thecircuit components are similar [].

In the next section, the equations or this type o con-guration will be derived with the spatial phase errors beingthe measurable state variables. Ten, in order to analyze thetracking mode o the system, a linear model is developed,resulting in two decoupled equations relating the spatialerrors to the delayed spatial errors.

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

constant o the dissipative terms. Te calculation methodor this critical value is presented, ollowed by numericalresults and simulations conrming the conclusions.

2. Triangular PLL Network

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

(1) ()+ 1(1) (1) ()= (1) sin 11(1) ()+ 12(2) 21

+ 13

(3)

31

(1)

() ,

5

0

5

10

15

20

25

30

35

T

n= 3

n= 2

n= 1

n= 0

1,2

(T

)

1(T)

2(T)

2 5/23 3/2 7/2

F :ollowing () and (). One curve or each.

(2) ()+ 1(2) (2) ()= (2) sin 22(2) ()+ 21(1) 12

+ 23(3) 32 (2) () ;(3) ()+ 1(3) (3) ()

= (3) sin 33(3) ()+ 31(1) 13

+ 32

(2)

23

(3)

() . ()At this point, it is assumed that the nodes have the same

constitutive parameters, that is, = (1) = (2) = (3) and = (1) = (2) = (3). Tese assumptions are reasonableconsidering a practical communication point o view, as thenodes are composed o standard blocks []. Besides, due tothe same motive, it is considered that all nodes have the sameaccuracy and, consequently, the phases o the three nodesare used to adjust the phase o the local node with the sameweight, resulting that,= 1/3, (, )[].

In order to be more accurate rom an engineering pointo view, the delays between the nodes ought to be different.

But in real modern networks, they assume low values, beingractions about10% o the period o the VCO oscillations,that is,10% o the node ree-running period. Tereore,here all the delays are considered to be equal, that is,,= , ( = ), and the results can be related to maximumpermitted delays or real networks [].

aking the ormer conditions into consideration, theequations o the phase nodes become

(1) ()+1(1) ()= sin 1

3(2) ( )+ (3) ( ) 2

3(1) () ,


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4 2 0 2 4 6 8 10 124

3

2

1

0

1

2

3

4104

104(t)

(t)

(a)

4 2 0 2 4 6 8 10 124

3

2

1

0

1

2

3

4104

104 (t)

(t)

(b)

1 0.5 0 0.5 1

8

6

4

2

0

2

4

6

8

10

104

103 (t)

(t)

(c)

F : Simulations or () with parameter = 2or three different values o. (a) < 1 ; initial condition:(1.0101, 0); attractor:equilibrium point. (b) = 1 ; initial condition:(1.0101, 0); attractor: limit-cycle. (c) > 1 ; initial condition:(1.0103, 0); attractor:innity.

(2) ()+1(2) ()

= sin 13(1) ( )+ (3) ( ) 23(2) ()(3) ()+1(3) ()= sin 13(1) ( )+ (2) ( ) 23(3) () .

()

o proceed with the analysis o the network, it mustbe considered that to obtain the node clocks in a correctsequence, the phase and requency errors between the nodes

have to vanish afer any kind o perturbation, with as mini-mum overshoot and tracking time as possible. Consequently,the important state variables are the spatial phase (

,) and

requency (,) errors, dened as

,= () (); (),= () (). ()

In this case, a three-node network, there are six indexcombinations (,) dening the spatial errors (,)and (,).However, i (1,2)and(1,3), combined with (1,2)and (1,3),are chosen, all the other spatial errors can be expressed as


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336 338 340 342 344 346 348 350 3522.5

2

1.5

1

0.5

0

0.5

1

1.5

2

2.5

Time t

w1= 2/T

104

(t)

F :() underFigure (b)conditions.linear combinations o them. Tereore, combining (), thesystem can be described by

1,2+11,2= sin 13(2) ( )+ (3) ( ) 23(1) ()

sin 13(1) ( )+ (3) ( ) 23(2) () ,()

1,3

+1

1,3

= sin 13(2) ( )+ (3) ( ) 23(1) () sin 13(1) ( )+ (2) ( ) 23(3) () .

()

3. Analysis of the Tracking Mode

An important indicator o the perormance o a clock dis-tribution systems is regarding its behavior in the trackingmode; that is, how the system recoversthe synchronism whena perturbation around the synchronous state occurs [].

In this case, the phase deviation can be considered to besmall and the argument o the sin unctions in ()and () canreplace the unctions []. Tereore, the equations describingthe system are

1,2+11,2+231,2131,2( )= 0; ()1,3+11,3+231,3131,3( )= 0. ()

Consequently, the model or the tracking mode o atriangular PLL network is given by two decoupled equations,which are ordinary, homogeneous, and linear, containinga delay term. Tis equation admits an analytical solution.

According to [, ], this type o delay differential equa-tion corresponds to a transcendental characteristic equation,decomposable as

=0

+ =0

= 0, ()which permits to calculate the critical stability value or thedelay.4. Critical Delay Value

Te critical delay value or the three-node clock distributionnetwork canbe determined considering that the spatialerrorsare described by

()+1 ()+23 13 ( )= 0, ()with parameter R

+

representing the time constant othe dissipative term and the biurcation parameter R+representing the signal processing and propagation delay.

In order to determine how the biurcation parameterdetermines the system stability it is necessary, or a given,to know the set R+ such that, or a given = ,solutions or () change stability.

For the dynamical system described by (), the charac-teristic equation is given by

(;,)= 2 ++2313 = 0 ()and can be written shortly as

(;,)= (; ) = 0, ()where(; ) = 32 + (3/) + 2.

It is supposed that there exists = , called critical , thatleads roots o the characteristic equation up to the imaginaryaxis, such that = with ( R+), implying the possibleonset o a Hop biurcation [].

Consequently, the module condition

;= = 1 ()must hold. Tereore, considering only positive values o

,

rom (), it is possible to nd

= () . ()Te number o values ordepends on the degree o the

polynomial(; ). Considering only the positive and realones, the rootshave to also satisy the angle condition or(), given by

tan1 I((;))

R ( (; ))= 2 = , ()or some

Z.


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Ten, rom () and (), it is possible to nd a relation-ship between a givenand its associated critical:()= 1 ()tan1I((;))R((;))=() 2. ()

By computing the module condition given by (), or(;), there are our possible roots: ()= 16242 3 44 242 + 9

1/2, ()and replacing () into () results in

()= 1 ()[tan1 3 () 2 3()2 2]. ()Equation () shows that, or a given, there are innite

periodical critical values or, with period:2 () . ()

It can be noticed that there are some situations with ()having no positive roots, meaning that, or some values o,there is nothat leads the roots up to the imaginary axis. Inthis case, system stability will depend on the position o theroots or = 0, or the given.5. Numerical Results and Critical Delays

For a system given by (), it is possible, by using (),to calculate which values o

allow critical time delays

, responsible or biurcations characterized by changingstability.Figure shows the real positive values o, given by ().

It can be seen that these roots suddenly appear or values ogreater than3/4. Analyzing characteristic equation (),or = 0, it can be seen that the system described by () isstable or every R+. Ten, or0 < < 3/4, the systemdescribed by () remains stable or every value o time delay.

However, considering a practical point o view, it isimportant to know how the critical delay values () arerelated to the dissipative time constant (). Equations ()and () show that, or a set o values o, there existamilies o possible

, depending on

(whenever

> 3/4),

corresponding in () to two real positive roots,or each given, each one associated with a critical , asshownin Figure .6. Engineering Results

Te results presented here enable the design o a clockdistribution network with second-order dissipative nodes []considering delays between nodes as a complement o thedesigning technics derived in [] or delay-ree networks.

Te general idea is to maintain the delays o the networksmaller than the critical value derived here. Under thiscondition, the reachability o the synchronous state ollowsthe criteria shown in [].

As the equations considered here are or normalizedangular requency and the time constant represents thenode lter, it must be equal to2, in order to avoid double-requency jitter [,].

Consequently, taking = 2 and using () result in1= 0.9804 rad/seg and2= 0.5889 rad/seg. From (),the corresponding critical delay values are

1

= 3.7013and2= 10.1853, or = 1.Simulations o () were perormed by using DDE

Matlabs subroutine, with integration step = 0.1 andsimulation time = 1000, or three different situations:Figure (a): < 1 , Figure (b): = 1 , andFigure (c): > 1 .

Finally, it can be concluded that1 is a biurcationparameter. Figure shows a periodic solution, related toFigure (b). Te angular requency experimentally measuredor()is1= 0.98 rad/seg.Conflict of Interests

Te authors declare that there is no conict o interestsregarding the publication o this paper.

References

[] W. C. Lindsey, F. Ghazvinian, W. C. Hagmann,and K. Dessouky,Network synchronization,Proceedings of the IEEE, vol. , no., pp. , .

[] iming Requirements o Slave Clocks Suitable or Use as NodeClocks in Synchronization NetworksRecomendation G.,IU-, .

[] H. Meyr and G. Ascheid,Synchronization in Digital Communi-

cations Phase-Frequency-Locked Loops, and Amplitude Control,vol. , John Wiley & Sons, New York, NY, USA, .

[] A. M. Bueno, A. A. Ferreira, andJ. R. C. Piqueira,Modelingandltering double-requency jitter in one-way masterslave chainnetworks,IEEE Transactions on Circuits and Systems I, vol. ,no. , pp. , .

[] A. Mehrotra, Noise analysis o phase-locked loops, IEEETransactions on Circuits and Systems I, vol. , no. , pp. , .

[] S. Bregni, Synchronization of Digital Telecommunications Net-works, John Wiley & Sons, New York, NY, USA, .

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Synchronous state in a ully connected phase-locked loopnetwork, Mathematical Problems in Engineering, vol. ,Article ID , pages, .

[] J. Stensby, False lock and biurcation in the phase locked loop,SIAM Journal on Applied Mathematics, vol. , no. , pp. , .

[] R. D. Cideciyan and W. C. Lindsey, Effects o long-term clockinstability on master-slave networks, IEEE Transactions onCommunications, vol. , no. , pp. , .

[] K. L. Cooke and Z. Grossman, Discretedelay, distributeddelayand stability switches, Journal of Mathematical Analysis andApplications, vol. , no. , pp. , .


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[] J. E. Forde, Delay differential equation models in mathemati-cal biology [Ph.D. thesis], Te University o Michigan, ,http://www.math.utah.edu/orde/research/JFthesis.pd.

[] J. R. C. Piqueira and L. H. A. Monteiro, All-pole phase-lockedloops: calculating lock-in range by using Evans root-locus,International Journalof Control, vol. , no. , pp. ,.

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