Limit cycles in bang-bang phase-locked loops (PDF Download
Limit Cycles in Bang-Bang Phase-Locked Loops Alexey Teplinsky
Raymond Flynn and Orla Feely
Institute of Mathematics National Academy of Science of Ukraine 3 Tereshchenkivska St, Kyiv 01601, Ukraine Email: [email protected]
School of Electrical, Electronic and Mechanical Engineering University College Dublin Belfield, Dublin 4, Ireland Email: [email protected], [email protected]
Abstract—This paper examines the nonlinear dynamics of a model of a second order Bang-Bang Phase-Locked Loop (BBPLL). Three distinct steady state dynamical patterns (locking, slew-rate limiting and limit cycles) have been observed for this discrete system. A corresponding continuous model of the BBPLL is established. This paper focuses on the occurrence and the shape of the limit cycles. In particular, equations for the limit cycle trajectories are determined. The condition for the appearance of limit cycles is then established as a boundary in parameter space. A further theorem transfers this analysis back to the discrete system, where a continuum of cycles is found to occur. A direct relationship between the level of input phase deviation and the occurrence of limit cycles is observed.
I.
INTRODUCTION
Phase-lock loops (PLLs) are closed-loop systems with negative feedback. The circuit essentially locks onto the frequency of an incoming signal and maintains this lock by extracting the phase error and aiming to reduce it to zero. The circuit is predominantly used in communications applications such as frequency synthesis and clock and data recovery. Recent research has seen significant attempts to better understand the complicated dynamics of PLLs through the application of nonlinear theory [1], [2]. In the field of Clock and Data Recovery (CDR) circuits, the Bang-Bang PLL (BB-PLL), Fig. 1, has emerged as an advantageous choice [3], [4]. This is mainly due to its high data rate capabilities and relative ease of implementation [5]. BB-PLLs, unlike standard PLLs, do not have a linear phase detector (PD) characteristic. The phase error output is ±1, depending on whether the input phase is leading or lagging the VCO phase. Unlike the linear PD, no attempt is made to measure the phase error. This will obviously increase the speed of the circuit, but at the cost of introducing further nonlinearity to the system. This paper examines the nonlinear dynamics exhibited by a model of the BB-PLL, Fig. 1. Our previous analysis [1] focused on the three distinct dynamical patterns exhibited by the system, which were locking, slew-rate limiting and limit cycles. A continuous model for the BB-PLL was developed, which allowed for boundaries of the dynamical patterns to be found. However we did not closely examine the structure of these dynamical patterns. In this paper we present a theorem for a class of limit cycles in the continuous model and give explicit equations for these limit cycles.
K
θd
Phase Detector
εn
∑ 1 sτ
θv
VCO
Figure 1. Block diagram of the BB-PLL
The conditions stated in the theorem are evaluated for the case of sinusoidal phase deviation [3], [5] and the region of existence of the limit cycles described by the theorem is calculated. A further theorem transfers this analysis back to the discrete system, where a continuum of cycles is observed, with similar equations describing the motion along the limit cycles. II.
CIRCUIT MODEL
A.
Original Model The second order BB-PLL to be examined in this work is described in [5], and is modelled by the following set of equations.
θ d (t n ) = θ d (0) + 2πt n δf + φ (t n )
θ v (t n +1 ) = θ v (t n ) + θ bb ε n +
εn 2 + ξ ξ
ε n = sgn[θ d (t n ) − θ v (t n )]
(1)
n
∑ε 0
n
(2) (3)
where θ d (t n ) represents the phase of the received data signal and θ v (t n ) represents the phase of the VCO output signal, at the nth sampling time t n . The time update equation is assumed to be uniform [5] and of the form t n +1 = t n + 1
f nom
(4)
where f nom is the centre frequency of the VCO. The frequency difference between the incoming data signal and the VCO is ∂f , with zero mean phase deviation of φ (t n )
This research is funded by Science Foundation Ireland (SFI).
0-7803-9390-2/06/$20.00 ©2006 IEEE
4074
ISCAS 2006
which has amplitude A and frequency f mod . The presence of φ (t n ) means the system is of third order. The terms θ bb and ξ are loop parameters. B. Change of Variables For our analysis, it is convenient to use the following substitutions u n = β (θ v (t n ) − θ d (0) )
vn = α
(5)
n −1
∑ε
n
(6)
θ n = f mod t n
(7)
i =0
2 and β = where α = ξ +3
1
A. Continuous System in the Half-Spaces We defined an autonomous continuous-time system in C simply by presenting a rule of how to produce a trajectory {(θ (t ), u (t ), v (t )) | t ≥ t 0 } for a given initial point (θ (t 0 ), u (t 0 ), v(t 0 )) ∈ C . First of all, the motion in angular variable θ is uniform with velocity ρ , so we put
θ (t ) = θ (t 0 ) + ρ (t − t 0 )
.
3
with jitter. In the case of slew-rate limiting the system switches between tracking the input signal, h(θn), and a slew motion. Finally, limit cycles correspond to trajectories moving around the origin with certain amplitude in a periodic manner. In order to find bounds for each type of behaviour, we approximated the discrete model (8)-(10) with a continuous system.
u (t ) = u (t 0 ) + (v (t 0 ) + α2 − 1)(t − t 0 ) − α2 (t − t 0 ) 2
This results in the following set of equations u n +1 = u n + v n − sgn (u n − h(θ n ) )
(8)
v n +1 = v n − α sgn (u n − h(θ n ) )
(9)
where u, v ∈ R (the real line) and θ ∈ S1 (the unit circle). For simplicity, we consider sinusoidal input h(θ n ) = β φ (t n ) = Aβ cos 2πθ n , with ∂f = 0 , but our approach is not dependent on this. The inclusion of the driving (or control) term
where ρ =
(10)
mod 1
f mod , completes our model as a discrete-time f nom
dynamical system in 3D space C = S 1 × R 2 . The convenience of our model, in comparison with the more usual approach of considering the phase error as the main variable, lies in the fact that the system (8)–(10) is not just piece-wise linear, but neither the input signal h(θ n ) nor the driving variable θ n affect the linear equations on the two appropriate pieces of phase space. Instead, they affect the shape of those pieces: that is, the critical manifold M h = {(θ , h(θ ), v)} decomposes the whole 3D space C into upper
C +h = {(θ , u, v) | u > h(θ )}
and
lower
C −h
= {(θ , u , v) | u < h(θ )} half-spaces on which the system equations (8)-(10) are linear and measure-preserving.
III.
(11)
Next, to be consistent with the discrete system, we derive the equations
θ bb 1 + ξ
θ n+1 = θ n + ρ
mod 1 .
DYNAMICAL PATTERNS
In a previous paper, [1], we identified three different dynamical patterns for the BB-PLL model in steady state. They were locking, slew-rate limiting and limit cycles. Locking corresponds to the system tracking the input signal
v(t ) = v(t 0 ) − α (t − t 0 )
(12)
inside C +h and u (t ) = u (t 0 ) + (v(t 0 ) − α2 + 1)(t − t 0 ) + α2 (t − t 0 ) 2 v(t ) = v(t 0 ) + α (t − t 0 )
(13)
inside C −h , so that the continuous trajectories follow the invariant curves of the discrete system. B. Continuous Sytem on the Critical Manifold In the discrete system, jitter of the output signal u around the input signal h(θ ) is characterized by the following property: if a point of trajectory is in C +h , then in the next step it is brought closer to C −h , and if it is in C −h , then in the next step it is brought closer to C +h . The value of v in this process oscillates around some value. This observation allows us to define the motion of a point on M h in the continuous system by the following simple rule: i) if the curve determined by the equations (11) and (12) moves into C +h , then the trajectory follows that curve; ii) if the curve determined by the equations (11) and (13) moves into C −h , then the trajectory follows that curve; iii) if neither i) nor ii) takes place, then the trajectory stays on M h with v = const until either i) or ii) happens (or stays for forever, if they never happen), and so the equations of its motion are
4075
u (t ) = h(θ (t 0 ) + ρ (t − t 0 )) v(t ) = v(t 0 )
.
(14)
C. Bounds of Dynamical Patterns The continuous system exhibits all the patterns of behaviour that correspond to the discrete patterns [1], but its dynamics are simpler and rigorous analysis is more readily applicable. The limit cycle pattern corresponds to true limit cycles in the continuous model. A necessary condition for their existence is Aβ ≥
1 α 1 − . 4ρ 2
IV.
1 α 1 − . 2πρ 2
(16)
LIMIT CYCLE THEOREMS
In order to further characterize the dynamical patterns, the following theorems further describe the behaviour of limit cycles, when the input h(θ) satisfies a certain set of conditions. The first theorem describes non-trivial (i.e., different from locking) limit cycles in the continuous system and the second transfers our analysis back to the original discrete system. A. Theorem 1 Suppose that a periodic input h(θ), with period normalised to one, satisfies the following condition: there 1 α exists θ sw ∈ S1 such that h(θ sw ) = 1 − and 4ρ 2 1 1 α hθ sw + = − for 1 − . Also h(θ ) < bup (θ ) 2 4ρ 2 1 θ ∈ θ sw ,θ sw + and h(θ ) > blow (θ ) for 2 1 θ ∈ θ sw + ,θ sw + 1 . 2
bup (θ ) = −
1 2α
α α α − (θ − θ sw ) − 1 − 2 4ρ ρ
and 1 2α
α α − + 4ρ ρ
2 1 α + 2α 4 ρ
2 2 α + 1 − 2
2 2 2 1 α 1 α α . + 1 − θ − θ sw − + 1 − − 2 2 2α 4 ρ 2
If these conditions are met, then there exists a cycle in the continuous system, whose projection in the (u,v)-plane is given by the following equations: For the upper branch u=−
1 2α
2 2 2 α 1 α α + 1 − (18) v + 1 − − 2 2α 4 ρ 2
α α with v ∈ − , . 4ρ 4ρ
Proof: Take an initial point θ (0) = θ sw , v(0) =
α , 4ρ
1 α 1 − and follow the definition of the 4ρ 2 continuous system to obtain the corresponding trajectory. It is found to traverse the parabola u (0) =
θ = θ sw + tρ mod 1 , v = until
it
reaches
the
α 4ρ
− tα , u = bup (θ sw + tρ )
point
1 1 = θ sw + mod 1 , 2 2ρ
θ
1 1 1 α α = − = − v , u 1 − at which point it 4ρ 2 4ρ 2ρ 2ρ switches to another parabola
θ = θ sw + tρ , v = −
α 4ρ
1 α u = blow (θ sw + tρ ) + t − 2 ρ
1 which directs the trajectory back to the point θ = θ sw , ρ 1 α 1 1 α v = , u = 1 − , completing the cycle in 2 ρ 4ρ ρ 4ρ the (u,v)-plane. These cycles are attracting, however the proof of this is too lengthy for this paper. Ñ
For our analysis, as in [3], [5], we take the case of the phase deviation modelled as a sinusoid, which gives h(θ ) = Aβ cos 2πθ and therefore to meet the conditions of 1 − α2 1 , arccos 2π 4 Aβρ which by itself implies (15). The criteria of the theorem can therefore be seen as conditions on Aβ and ρ, as α will be fixed under normal loop operating conditions, whereas the amplitude and frequency of the phase deviation (A and ρ) may vary greatly. The boundary on the parameter values that satisfy Theorem 1 is obtained in Subsection C.
Theorem 1, it is a requirement that θ sw =
Where:
blow (θ ) =
u=
(15)
The locking pattern corresponds to a lock without jitter in the continuous model, the case when a trajectory stays forever on the critical surface M h with constant v. The exact condition for this pattern to take place (at v = 0 ) is Aβ ≤
and for the lower branch
1 2α
2 2 2 1 α α α + 1 − (17) v − 1 − + 2 2α 4 ρ 2
B. Theorem 2 For the sinusoidal input case, if the condition of Theorem 1 1 is satisfied and ρ = with integer n ≥ 1 , then for any 4n given θ * ∈ S1 there exists a continuum of cycles of length
4076
4n in the discrete system over the values of θ which differ from θ* by kρ with integer k. Proof:
Observe
θ i = θ 0 + iρ mod 1 vi =
α α i − ⋅ 4 ρ ρ 4n
that the trajectory given by for all i, u i = bup (θ sw + iρ ) , 0 ≤ i ≤ 2n − 1
for
and
1 α α i − 2n u i = blow θ sw + + iρ , for vi = − + ⋅ 4 ρ ρ 4n 2 2n ≤ i ≤ 4n − 1 , is a 4n-cycle for any θ 0 ∈ (θ sw ,θ sw + ρ ] .
To prove this, note that the branches (17) and (18) are still invariant for the discrete system (in the same sense that they are for the continuous system), and that the following inequalities hold (due to the condition of Theorem 1): u i > h(θ i ) for 0 ≤ i ≤ 2n − 1 and u i < h(θ i ) for 2n ≤ i ≤ 4n − 1 . Finally, if the whole trajectory is shifted slightly along the u-axis, it is still maintained, so there are continuums of cycles as stated. Ñ C. Boundary on the Parameter Values The conditions specified by Theorem 1 can be represented graphically, as depicted in Fig. 2. For our analysis we consider the case of a sinusoidal h(θ), which by symmetry means only the (θsw, θsw+0.5) range of θ must be checked for compliance. Clearly, the inequality ∂h(θ sw )
∂θ
4 ρ 2πρ 8πρ 2 8π 2 ρ 3
)
(20)
V.
CONCLUSION
This paper focussed on the structure of the dynamical patterns of BB-PLLs. In particular, a theorem was presented that described a class of limit cycles for a continuous model of the system. Another theorem was presented, which extended the results to the discrete system case. The parameter boundary for this class was calculated using a condition observed from graphical analyses. A parameter plane plot illustrated the different regions of dynamical patterns depending on the magnitude and frequency of the input phase deviation. REFERENCES [1]
[2]
[3]
[4]
Figure 2. The (h,θ) plane (α = 0.002, Aβ = 1.5, ρ = 0.2) depicting sinusoidal h(θ) and the upper boundary bup(θ) over the range (θsw, θsw+0.5)
[5]
4077
A. Teplinsky, R. Flynn, and O. Feely, “Dynamical patterns of bangbang phase-locked loops,” proceedings of the 17th European Conference on Circuit Theory and Design (ECCTD), Cork, Ireland, vol. III, August 2005, pp. 405-408. G.M. Bernstein, M.A. Lieberman, and A.J. Lichtenberg, “Nonlinear dynamics of a digital phase-locked loops,” IEEE Trans. Commun., vol. 37, no. 10, October 1989, pp. 1062–1070. J. Lee, K.S. Kundert, and B. Razavi, “Analysis and modeling of bangbang clock and data recovery circuits,” IEEE Jour. of Solid State Circuits, vol. 39, no. 9, Sept. 2004, pp. 1571–1580. N. Da Dalt, “A design-oriented study of the nonlinear dynamics of digital bang-bang PLLs,” IEEE Trans. Circuits and Systems-I, vol. 52, no. 1, January 2005, pp. 21-31. R.C. Walker, “Designing bang-bang PLLs for clock and data recovery in serial data transmission systems,” in Phase-Locking in High-Performance Systems, B Razavi, Ed: IEEE Press, 2003, pp. 3445.
Raymond Flynn and Orla Feely
Institute of Mathematics National Academy of Science of Ukraine 3 Tereshchenkivska St, Kyiv 01601, Ukraine Email: [email protected]
School of Electrical, Electronic and Mechanical Engineering University College Dublin Belfield, Dublin 4, Ireland Email: [email protected], [email protected]
Abstract—This paper examines the nonlinear dynamics of a model of a second order Bang-Bang Phase-Locked Loop (BBPLL). Three distinct steady state dynamical patterns (locking, slew-rate limiting and limit cycles) have been observed for this discrete system. A corresponding continuous model of the BBPLL is established. This paper focuses on the occurrence and the shape of the limit cycles. In particular, equations for the limit cycle trajectories are determined. The condition for the appearance of limit cycles is then established as a boundary in parameter space. A further theorem transfers this analysis back to the discrete system, where a continuum of cycles is found to occur. A direct relationship between the level of input phase deviation and the occurrence of limit cycles is observed.
I.
INTRODUCTION
Phase-lock loops (PLLs) are closed-loop systems with negative feedback. The circuit essentially locks onto the frequency of an incoming signal and maintains this lock by extracting the phase error and aiming to reduce it to zero. The circuit is predominantly used in communications applications such as frequency synthesis and clock and data recovery. Recent research has seen significant attempts to better understand the complicated dynamics of PLLs through the application of nonlinear theory [1], [2]. In the field of Clock and Data Recovery (CDR) circuits, the Bang-Bang PLL (BB-PLL), Fig. 1, has emerged as an advantageous choice [3], [4]. This is mainly due to its high data rate capabilities and relative ease of implementation [5]. BB-PLLs, unlike standard PLLs, do not have a linear phase detector (PD) characteristic. The phase error output is ±1, depending on whether the input phase is leading or lagging the VCO phase. Unlike the linear PD, no attempt is made to measure the phase error. This will obviously increase the speed of the circuit, but at the cost of introducing further nonlinearity to the system. This paper examines the nonlinear dynamics exhibited by a model of the BB-PLL, Fig. 1. Our previous analysis [1] focused on the three distinct dynamical patterns exhibited by the system, which were locking, slew-rate limiting and limit cycles. A continuous model for the BB-PLL was developed, which allowed for boundaries of the dynamical patterns to be found. However we did not closely examine the structure of these dynamical patterns. In this paper we present a theorem for a class of limit cycles in the continuous model and give explicit equations for these limit cycles.
K
θd
Phase Detector
εn
∑ 1 sτ
θv
VCO
Figure 1. Block diagram of the BB-PLL
The conditions stated in the theorem are evaluated for the case of sinusoidal phase deviation [3], [5] and the region of existence of the limit cycles described by the theorem is calculated. A further theorem transfers this analysis back to the discrete system, where a continuum of cycles is observed, with similar equations describing the motion along the limit cycles. II.
CIRCUIT MODEL
A.
Original Model The second order BB-PLL to be examined in this work is described in [5], and is modelled by the following set of equations.
θ d (t n ) = θ d (0) + 2πt n δf + φ (t n )
θ v (t n +1 ) = θ v (t n ) + θ bb ε n +
εn 2 + ξ ξ
ε n = sgn[θ d (t n ) − θ v (t n )]
(1)
n
∑ε 0
n
(2) (3)
where θ d (t n ) represents the phase of the received data signal and θ v (t n ) represents the phase of the VCO output signal, at the nth sampling time t n . The time update equation is assumed to be uniform [5] and of the form t n +1 = t n + 1
f nom
(4)
where f nom is the centre frequency of the VCO. The frequency difference between the incoming data signal and the VCO is ∂f , with zero mean phase deviation of φ (t n )
This research is funded by Science Foundation Ireland (SFI).
0-7803-9390-2/06/$20.00 ©2006 IEEE
4074
ISCAS 2006
which has amplitude A and frequency f mod . The presence of φ (t n ) means the system is of third order. The terms θ bb and ξ are loop parameters. B. Change of Variables For our analysis, it is convenient to use the following substitutions u n = β (θ v (t n ) − θ d (0) )
vn = α
(5)
n −1
∑ε
n
(6)
θ n = f mod t n
(7)
i =0
2 and β = where α = ξ +3
1
A. Continuous System in the Half-Spaces We defined an autonomous continuous-time system in C simply by presenting a rule of how to produce a trajectory {(θ (t ), u (t ), v (t )) | t ≥ t 0 } for a given initial point (θ (t 0 ), u (t 0 ), v(t 0 )) ∈ C . First of all, the motion in angular variable θ is uniform with velocity ρ , so we put
θ (t ) = θ (t 0 ) + ρ (t − t 0 )
.
3
with jitter. In the case of slew-rate limiting the system switches between tracking the input signal, h(θn), and a slew motion. Finally, limit cycles correspond to trajectories moving around the origin with certain amplitude in a periodic manner. In order to find bounds for each type of behaviour, we approximated the discrete model (8)-(10) with a continuous system.
u (t ) = u (t 0 ) + (v (t 0 ) + α2 − 1)(t − t 0 ) − α2 (t − t 0 ) 2
This results in the following set of equations u n +1 = u n + v n − sgn (u n − h(θ n ) )
(8)
v n +1 = v n − α sgn (u n − h(θ n ) )
(9)
where u, v ∈ R (the real line) and θ ∈ S1 (the unit circle). For simplicity, we consider sinusoidal input h(θ n ) = β φ (t n ) = Aβ cos 2πθ n , with ∂f = 0 , but our approach is not dependent on this. The inclusion of the driving (or control) term
where ρ =
(10)
mod 1
f mod , completes our model as a discrete-time f nom
dynamical system in 3D space C = S 1 × R 2 . The convenience of our model, in comparison with the more usual approach of considering the phase error as the main variable, lies in the fact that the system (8)–(10) is not just piece-wise linear, but neither the input signal h(θ n ) nor the driving variable θ n affect the linear equations on the two appropriate pieces of phase space. Instead, they affect the shape of those pieces: that is, the critical manifold M h = {(θ , h(θ ), v)} decomposes the whole 3D space C into upper
C +h = {(θ , u, v) | u > h(θ )}
and
lower
C −h
= {(θ , u , v) | u < h(θ )} half-spaces on which the system equations (8)-(10) are linear and measure-preserving.
III.
(11)
Next, to be consistent with the discrete system, we derive the equations
θ bb 1 + ξ
θ n+1 = θ n + ρ
mod 1 .
DYNAMICAL PATTERNS
In a previous paper, [1], we identified three different dynamical patterns for the BB-PLL model in steady state. They were locking, slew-rate limiting and limit cycles. Locking corresponds to the system tracking the input signal
v(t ) = v(t 0 ) − α (t − t 0 )
(12)
inside C +h and u (t ) = u (t 0 ) + (v(t 0 ) − α2 + 1)(t − t 0 ) + α2 (t − t 0 ) 2 v(t ) = v(t 0 ) + α (t − t 0 )
(13)
inside C −h , so that the continuous trajectories follow the invariant curves of the discrete system. B. Continuous Sytem on the Critical Manifold In the discrete system, jitter of the output signal u around the input signal h(θ ) is characterized by the following property: if a point of trajectory is in C +h , then in the next step it is brought closer to C −h , and if it is in C −h , then in the next step it is brought closer to C +h . The value of v in this process oscillates around some value. This observation allows us to define the motion of a point on M h in the continuous system by the following simple rule: i) if the curve determined by the equations (11) and (12) moves into C +h , then the trajectory follows that curve; ii) if the curve determined by the equations (11) and (13) moves into C −h , then the trajectory follows that curve; iii) if neither i) nor ii) takes place, then the trajectory stays on M h with v = const until either i) or ii) happens (or stays for forever, if they never happen), and so the equations of its motion are
4075
u (t ) = h(θ (t 0 ) + ρ (t − t 0 )) v(t ) = v(t 0 )
.
(14)
C. Bounds of Dynamical Patterns The continuous system exhibits all the patterns of behaviour that correspond to the discrete patterns [1], but its dynamics are simpler and rigorous analysis is more readily applicable. The limit cycle pattern corresponds to true limit cycles in the continuous model. A necessary condition for their existence is Aβ ≥
1 α 1 − . 4ρ 2
IV.
1 α 1 − . 2πρ 2
(16)
LIMIT CYCLE THEOREMS
In order to further characterize the dynamical patterns, the following theorems further describe the behaviour of limit cycles, when the input h(θ) satisfies a certain set of conditions. The first theorem describes non-trivial (i.e., different from locking) limit cycles in the continuous system and the second transfers our analysis back to the original discrete system. A. Theorem 1 Suppose that a periodic input h(θ), with period normalised to one, satisfies the following condition: there 1 α exists θ sw ∈ S1 such that h(θ sw ) = 1 − and 4ρ 2 1 1 α hθ sw + = − for 1 − . Also h(θ ) < bup (θ ) 2 4ρ 2 1 θ ∈ θ sw ,θ sw + and h(θ ) > blow (θ ) for 2 1 θ ∈ θ sw + ,θ sw + 1 . 2
bup (θ ) = −
1 2α
α α α − (θ − θ sw ) − 1 − 2 4ρ ρ
and 1 2α
α α − + 4ρ ρ
2 1 α + 2α 4 ρ
2 2 α + 1 − 2
2 2 2 1 α 1 α α . + 1 − θ − θ sw − + 1 − − 2 2 2α 4 ρ 2
If these conditions are met, then there exists a cycle in the continuous system, whose projection in the (u,v)-plane is given by the following equations: For the upper branch u=−
1 2α
2 2 2 α 1 α α + 1 − (18) v + 1 − − 2 2α 4 ρ 2
α α with v ∈ − , . 4ρ 4ρ
Proof: Take an initial point θ (0) = θ sw , v(0) =
α , 4ρ
1 α 1 − and follow the definition of the 4ρ 2 continuous system to obtain the corresponding trajectory. It is found to traverse the parabola u (0) =
θ = θ sw + tρ mod 1 , v = until
it
reaches
the
α 4ρ
− tα , u = bup (θ sw + tρ )
point
1 1 = θ sw + mod 1 , 2 2ρ
θ
1 1 1 α α = − = − v , u 1 − at which point it 4ρ 2 4ρ 2ρ 2ρ switches to another parabola
θ = θ sw + tρ , v = −
α 4ρ
1 α u = blow (θ sw + tρ ) + t − 2 ρ
1 which directs the trajectory back to the point θ = θ sw , ρ 1 α 1 1 α v = , u = 1 − , completing the cycle in 2 ρ 4ρ ρ 4ρ the (u,v)-plane. These cycles are attracting, however the proof of this is too lengthy for this paper. Ñ
For our analysis, as in [3], [5], we take the case of the phase deviation modelled as a sinusoid, which gives h(θ ) = Aβ cos 2πθ and therefore to meet the conditions of 1 − α2 1 , arccos 2π 4 Aβρ which by itself implies (15). The criteria of the theorem can therefore be seen as conditions on Aβ and ρ, as α will be fixed under normal loop operating conditions, whereas the amplitude and frequency of the phase deviation (A and ρ) may vary greatly. The boundary on the parameter values that satisfy Theorem 1 is obtained in Subsection C.
Theorem 1, it is a requirement that θ sw =
Where:
blow (θ ) =
u=
(15)
The locking pattern corresponds to a lock without jitter in the continuous model, the case when a trajectory stays forever on the critical surface M h with constant v. The exact condition for this pattern to take place (at v = 0 ) is Aβ ≤
and for the lower branch
1 2α
2 2 2 1 α α α + 1 − (17) v − 1 − + 2 2α 4 ρ 2
B. Theorem 2 For the sinusoidal input case, if the condition of Theorem 1 1 is satisfied and ρ = with integer n ≥ 1 , then for any 4n given θ * ∈ S1 there exists a continuum of cycles of length
4076
4n in the discrete system over the values of θ which differ from θ* by kρ with integer k. Proof:
Observe
θ i = θ 0 + iρ mod 1 vi =
α α i − ⋅ 4 ρ ρ 4n
that the trajectory given by for all i, u i = bup (θ sw + iρ ) , 0 ≤ i ≤ 2n − 1
for
and
1 α α i − 2n u i = blow θ sw + + iρ , for vi = − + ⋅ 4 ρ ρ 4n 2 2n ≤ i ≤ 4n − 1 , is a 4n-cycle for any θ 0 ∈ (θ sw ,θ sw + ρ ] .
To prove this, note that the branches (17) and (18) are still invariant for the discrete system (in the same sense that they are for the continuous system), and that the following inequalities hold (due to the condition of Theorem 1): u i > h(θ i ) for 0 ≤ i ≤ 2n − 1 and u i < h(θ i ) for 2n ≤ i ≤ 4n − 1 . Finally, if the whole trajectory is shifted slightly along the u-axis, it is still maintained, so there are continuums of cycles as stated. Ñ C. Boundary on the Parameter Values The conditions specified by Theorem 1 can be represented graphically, as depicted in Fig. 2. For our analysis we consider the case of a sinusoidal h(θ), which by symmetry means only the (θsw, θsw+0.5) range of θ must be checked for compliance. Clearly, the inequality ∂h(θ sw )
∂θ
4 ρ 2πρ 8πρ 2 8π 2 ρ 3
)
(20)
V.
CONCLUSION
This paper focussed on the structure of the dynamical patterns of BB-PLLs. In particular, a theorem was presented that described a class of limit cycles for a continuous model of the system. Another theorem was presented, which extended the results to the discrete system case. The parameter boundary for this class was calculated using a condition observed from graphical analyses. A parameter plane plot illustrated the different regions of dynamical patterns depending on the magnitude and frequency of the input phase deviation. REFERENCES [1]
[2]
[3]
[4]
Figure 2. The (h,θ) plane (α = 0.002, Aβ = 1.5, ρ = 0.2) depicting sinusoidal h(θ) and the upper boundary bup(θ) over the range (θsw, θsw+0.5)
[5]
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