PLL networks 1 Theoretical hypothesis

PLL networks 1

Theoretical hypothesis

The idea of this project is to study phase-locked loop (PLL) networks trying to relate node parameters and connection measures with possible behaviors of the whole network, exploring the several situations of synchronism and bifurcations. There are a lot of studies for network synchronization but with strong limitations either in the isolated node dynamics or in the nature of the connection. As the work proposition has no specific purpose, the development of the research will be free concerning node dynamics and topological arrangements. The starting point will be to consider that the nodes are second order PLL, i.e., one step ahead of the Kuramoto’s models where the dynamics of the isolated nodes are of first order. Since proposed by Bellescize in 1932, the phase-locked loop evolved from being a device mounted with discrete electronic components, to being applied in synchronous frequency demodulation systems [1] in different forms of integrated circuits and software implementations, which are present on computational systems, optics communications devices and even in smart-gride applications [2, 3]. Clearly, these different applications correspond to different frequency ranges and synchronism accuracy. However, considering that the PLL architecture is always the same, being described by a closed loop composed of phase-detector (PD), a low-pass filter (F) and a voltage controlled oscillator (VCO), represented in Fig.1, there are the following possible adjustments to be considered according to the application [4]: • PD gain and its linearity; • F frequency response; • VCO hold-in range.

Figure 1: PLL block diagram

1

The PD is supposed to be nonlinear with the output signal, vd (t), depending on the sine of the phase difference between the phase of the remote signal, present in the PD input, vi (t), and the phase of the local oscillation generated by the VCO, vo (t), since then non small phase errors must be taken into account [5, 6]. The filter that integrates the output of the PD will be considered a firstorder low-pass lag-lead. The output of the filter, vc (t), controls the VCO, which its derivative phase is proportional to the filter output [2]. In order to give the results in a similar way of the classical PLL literature [2], the transfer function of the filter is considered to be: R2 Cs + 1 , (1) (R1 + R2 )Cs + 1 with R1 and R2 having electrical resistance unit and C, electrical capacitance. 1 , To simplify the notation, some parameters can be defined: µ1 = (R1 +R 2 )C µ2 = K0 Kd Vi and µ3 = R2 C. It can be noticed that µ1 and µ2 have frequency units and µ3 is expressed in time units. Under these conditions [7], the PLL dynamics is expressed by: F (s) =

φ¨ + µ1 (1 + µ2 µ3 cos φ)φ˙ + µ1 µ2 sin φ = θ¨i + µ1 θ˙i ,

(2)

with φ representing the phase difference between signals vi (t) and vo (t), and θi , the phase of vi (t). To normalize the equations, the time t can be replaced by τ = µ1 t. Consequently, the dynamics of the second order PLL, for time measured by τ , is given by: ′′

′

′′

′

φ + (1 + c cos φ)φ + G sin φ = θi + θi ,

(3)

with the quotation marks representing the derivatives related to τ , c = µ2 µ3 and G = µµ21 . The first task of the project is to obtain all the possible behaviors of (3) for the several possible θi : steps, ramps, parabola, periodic functions and stochastic inputs.

2

Exploring behaviors

For different inputs, the possible behaviors are explored by using analytical results and simulations performed with a SIMULINK model, built in MATLAB R2013a. As the main function of the PLL is to synchronize the phase of the VCO output (θo ) with the phase of the PD input (θi ), the dynamics of the phase error φ is analyzed, for different phase inputs θi .

2.1

Phase step

A typical PLL phase input is the step, applied at t = 0+ . In this case, equation (3) is reduced to: 2

1.5

2.5

1 2

0.5 1.5

phase error

phase derivative

0

−0.5

1

−1 0.5

−1.5 0

−2

−2.5 −0.5

0

0.5

1 phase

1.5

2

−0.5

2.5

0

20

(a) Phase portrait

40

60 80 normalized time

100

120

140

(b) Temporal evolution

Figure 2: Autonomous case: Asymptotically stable equilibrium 0.5

2.5

0

2

−0.5 phase derivative

phase derivative

1.5 −1

−1.5

1

0.5 −2 0

−2.5

−3 −0.5

0

0.5

1

1.5 phase

2

2.5

3

3.5

−0.5

3

(a) Initial condition < π

3.5

4

4.5

5 phase

5.5

6

6.5

7

(b) Initial condition > π

Figure 3: Autonomous case: simulation near the saddle point

′′

′

φ + (1 + c cos φ)φ + G sin φ = 0.

(4)

The dynamics of equation (4) presents cylindrical phase surface and, in the interval ] − π, π], there are two equilibrium solutions: an asymptotically stable, ′ ′ (φ, φ ) = (0, 0), and a saddle, (φ, φ ) = (π, 0) [7], for any c ≥ 0 and G > 0. Figure 2 shows the simulation considering c = 1 and G = 4, with 2a representing the phase portrait and 2b, the temporal evolution of the phase error. It can be seen that the asymptotically stable equilibrium solution is reached. Simulation near the saddle point depends on the initial condition. For initial phase slightly lower than π, the solution tends asymptotically to the phase error φ = 0 (Fig 3a). Starting in a condition slightly greater than π, the solution tends asymptotically to the phase error φ = 2π (Fig 3b), i.e., a cycle slip appears. Consequently, it can be concluded that, for any phase step and any initial condition, the PLL provides synchronization with phase error asymptotically tending to zero or 2nπ (n cycle slips).

3

3.5

7

3

6

2.5 5 phase error

phase derivative

2 1.5 1

4

3

0.5 2 0 1

−0.5 −1

0

1

2

3

4

5

6

7

0

0

10

20

phase

(a) Phase portrait

30 normalized time

40

50

60

(b) Temporal evolution

Figure 4: Phase ramp with Ω > G

2.2

Phase ramp

There are some modulation process where the phase varies in a constant rate, i.e., the input signal presents a constant value for the frequency, here denoted by Ω. Under these conditions, the phase input is given by: θi = Ωt + θ0 ,

(5)

and, consequently, equation (3) takes the form: ′′

′

φ + (1 + c cos φ)φ + G sin φ = Ω.

(6)

The dynamics of equation (6) presents cylindrical phase surface and, in the interval ] − π, π]. There are tree different situations that depend on the relation between the loop gain G, and the frequency of the input phase Ω: • Ω > G, i.e., the frequency of the input signal greater than the loop gain; • Ω = G, i.e., the frequency of the input signal equal to the loop gain; • Ω < G, i.e., the frequency of the input signal lower than the loop gain. Considering the first case (Ω > G), with c = 1, G = 2 and Ω = 2.5, the system was simulated. Fig. 4a shows the phase portrait and Fig. 4b, the temporal evolution of the phase error. In the both figures, the phase is represented with modulus 2π. It can be seem that there is no equilibrium solution for the phase error, because its derivative is different from zero almost everywhere. In engineering language, the value of the frequency of the input signal (Ω) is out of the capture range. In the second case (Ω = G), there is one equilibrium point the interval ′ ] − π, π], given by (φ, φ ) = (π/2, 0), which is non hyperbolic and, consequently, Ω = G is a bifurcation parameter [8]. Simulating this situation with c = 1

4

0.4

2

0.2

1.8

0 1.6 phase error

phase derivative

−0.2 −0.4 −0.6

1.4

1.2

−0.8 1 −1 0.8

−1.2 −1.4

0.8

1

1.2

1.4

1.6

1.8

2

0

10

20

phase

(a) Phase portrait

30 normalized time

40

50

60

(b) Temporal evolution

Figure 5: Phase ramp with Ω = G 1

2 1.8

0.5

1.6 1.4 phase error

phase derivative

0

−0.5

−1

1.2 1 0.8 0.6 0.4

−1.5

0.2 −2

0

0.5

1 phase

1.5

(a) Phase portrait

2

0

0

10

20

30 normalized time

40

50

60

(b) Temporal evolution

Figure 6: Phase ramp with Ω < G (Asymptotically stable equilibrium) and G = Ω = 2, Fig. 5a shows the phase portrait and Fig. 5b, the temporal evolution of the phase error. Observing the simulation, it can be concluded that, under these condition, ′ the equilibrium point (φ, φ ) = (π/2, 0) is asymptotically stable. Concerning to the third case (Ω < G), the dynamics of equation (6) presents cylindrical phase surface and, in the interval ] − π, π], there are two equilibrium ′ ′ solutions: an asymptotically stable, (φ, φ ) = (φ1 , 0), and a saddle, (φ, φ ) = (φ2 , 0) [7], such that: Ω sin φ1 = sin φ2 = ; cos φ1 = − cos φ2 = G

r

1−(

Ω 2 ) . G

Figure 6 shows the simulation considering c = 1, G = 4, and Ω = 2 with 6a representing the phase portrait and 6b, the temporal evolution of the phase error. It can be seen that the asymptotically stable equilibrium solution is reached. For the parameters used in this example of the case ω < G, the asymptotically stable stable point, in the interval ] − π, π], corresponds to φ1 = π6 , a fact that is clear observing figures 6a and 6b. Consequently, φ2 = 5π 6 is a saddle 5

0.5

3.5

3

0

2.5 phase error

phase derivative

−0.5

−1

2

1.5

−1.5 1 −2

−2.5

0.5

0

0.5

1

1.5 phase

2

2.5

(a) Phase portrait

3

0

0

10

20

30 normalized time

40

50

60

(b) Temporal evolution

Figure 7: Phase ramp with Ω < G (Saddle point) point for the system. For initial phase slightly different from to the phase error φ1 , as Fig 7 shows.

6

5π 6 ,

the solution tends asymptotically

References [1] H. de Bellescize, “La reception synchrone”, Onde Electrique, vol 11, pp. 230-240, 1932. [2] R. E. Best, Phase-Locked Loops, 6th ed. New York: McGraw Hill, 2007. [3] N. R. N. Ama, F. O. Martinz, L. Matakas Jr. & F. Kassab Jr., “Phaselocked loop based on selective harmonics elimination for utility applications”, IEEE Transactions on Power Electronics, vol 28(1), pp. 144-153, 2013. [4] A. M. Bueno, A. G. Rigon, A. A. Ferreira & J. R. C. Piqueira, “Design constraints for third-order PLL nodes in master-slave clock distribution networks”, Communications in Nonlinear Science and Numerical Simulation, vol 15(9), pp. 2565-2574, 2010. [5] G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev & R. V. Yuldashev, “Nonlinear dynamical model of Costas loop and an approach to the analysis of its stability in the large”, Signal Processing, vol 108, pp. 124-135, 2015. [6] J. R. C. Piqueira, “Using bifurcations in the determination of lock-in ranges for third-order phase-locked loops”, Communications in Nonlinear Science and Numerical Simulation, vol 14(5), pp. 2328-2335, 2009. [7] J. R. C. Piqueira, Aplica¸c˜ ao da Teoria Qualitativa de Equa¸c˜ oes Diferenciais a Problemas de Sincronismo de Fase, Doctoral Thesis, Escola Polit´ecnica da Universidade de S˜ ao Paulo: S˜ ao Paulo, 1987. [8] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, New York: Springer-Verlag, 1995. [9] Yi-Bo Zhaoa, Du-Qu Weib & Xiao-Shu Luo, “Study on chaos control of second-order non-autonomous phase-locked loop based on state observer”, Chaos, Solitons & Fractals, 39(4), pp. 1817-1822, 2009. [10] B. A. Harb & A. M. Harb, “Chaos and bifurcation in a third-order phase locked loop”, Chaos, Solitons & Fractals, 19, pp. 667-672, 2004. [11] A. M. Harb & B. A. Harb, “Chaos control of third-order phase-locked loops using backstepping nonlinear controller”, Chaos, Solitons & Fractals, 20, pp. 719-723, 2004. [12] R. B. Pinheiro & J. R. C. Piqueira, “Designing All-Pole Filters for HighFrequency Phase-Locked Loops”, Mathematical Problems in Engineering, 2014, ID 682318, pp. 1-8, 2014. [13] J. R. C. Piqueira & L. H. A. Monteiro, “All-pole phase-locked loops: calculating lock-in range by using Evan’s root-locus”, International Journal of Control, 79(7), pp. 822-829, 2006. 7

[14] G. A. Leonov, N. V. Kuznetsov, M. V. Yuldashev & R. V. Yuldashev, “Hold-In, Pull-In, and Lock-In Ranges of PLL Circuits: Rigorous Mathematical Definitions and Limitations of Classical Theory”, IEEE Transactions on Circuits and Systems I: Regular Papers, 62(10), pp. 2454-2464, 2015. [15] G.A. Leonov, N.V. Kuznetsov, M.V. Yuldashev & R.V. Yuldashev, “Mathematical Models of the Costas Loop”, Doklady Mathematics, 92(2), pp. 594-598, 2015. [16] J. E. Marsden & M. McCracken, Hopf Bifurcation and its Applications, New York: Springer-Verlag, 1976. [17] K. Ogata, Modern Control Engineering, 5th ed. New Jersey: Prentice Hall, 2011. [18] A. B. Poore, “On the theory and applications of Hopf-Friedrichs bifurcation”, Archives of Rational Mechanics and Analysis, vol 60, pp. 371-393, 1976. [19] A. J. Lichtenberg & M. A. Liebernman, Regular and Chaotic Dynamics, 2nd ed. New York: Springer-Verlag, 1992.

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