Theory of the Non-linear Analog Phase Locked Loop

314 Pages·2004·9.4 MB·English

by  Nikolaos I. Margaris

Lecture Notes
in Control and Information Sciences 304
Editors:M.Thoma · M.Morari
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Nikolaos I. Margaris
Theory of the
Non-linear Analog
Phase Locked Loop
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A.Bensoussan·P.Fleming·M.J.Grimble·P.Kokotovic·
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Author
Prof.NikolaosI.Margaris
AristotleUniversityofThessaloniki
DepartmentofElectricalandComputerEngineering
54124Thessaloniki
Greece
ISSN0170-8643
ISBN3-540-21339-2 Springer-VerlagBerlinHeidelbergNewYork
LibraryofCongressControlNumber:2004104484
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To
Lena, Ioannis and Konstantinos
Preface
Φύσιςκρύπτεσθαιφιλει(cid:792)
‛Ηράκλειτος*
This book develops for the first time a complete and connected non-linear
theory for the analog Phase-Locked Loop (PLL), which clarifies the ob-
scure points of its complex non-linear behaviour.
The material of the book is divided into six parts. The extensive and te-
dious mathematic calculations and the experimental results are presented in
ten appendices included in the sixth part.
Part I. Preliminaries: Chapter 1 gives a brief description of the differ-
ent kinds of PLL and reveals the weak points of the conventional PLL the-
ory. These weak points arise from the dynamics of the controlled oscilla-
tor (VCO) and the ripple produced by the phase detector (PD), which are
ignored in the conventional PLL theory.
Chapter 2 gives the description of the loop components (controlled os-
cillator, phase detector and low-pass filter) taking into account the satura-
tion and the time delays that appear in the forward and the feedback paths
of the loop. The controlled oscillator is represented by a parametric version
of Liénard’s equation. Liénard's equation is an almost complete non-linear
model of the oscillators dominant dynamics.
Part II. First order PLL: Chapter 3 deals with the general behaviour
of the first order PLL. Even though the first order loop is not of great prac-
tical importance, its investigation provides the physical interpretation of
the PLL mechanism and gives considerable insight into the nonlinear be-
haviour of higher order loops. It is proven that synchronization is achieved
through two actions: a control (regulation) and a parametric action, caused
* Nature enjoys hiding. Heraclitos
VIII Preface
by the ripple. In addition, it is proven that any time delay appearing in the
forwardpathoftheloopdoesnotaffectthePLLperformance.
Chapter4analyzesthemainsynchronizationofthefirstorderPLL.The
analysis results in closed form expressions for the hold-in range, the track-
ing range, the lock-in time and the beat-note frequency. It is proved that
the time delay in the feedback path causes a significant reduction of the
main hold-in range. Furthermore, it is shown that the ripple causes an un-
desirable parametric resonance when the controlled oscillator is an almost
harmonic oscillator. This resonance leads the PLL elements to saturation
anddestroystheanalogbehaviouroftheloop.
Chapter 5 discusses the synchronization of the first order loop at the
thirdharmonicofthecenterVCOfrequency.Itisproventhattheharmonic
synchronization is due to a parametric action of the ripple and it is inde-
pendent of time delay. In addition, it is proved that in the absence of time
delay theharmonichold-inrangeisequaltothemainhold-inrange. How-
ever, in the presence of time delay the harmonic hold-in range is greater
than the main hold-in range. This means that the harmonic (false) locking
probabilityinthefirstorderPLLishigh.
Part III. Second order type-I PLL: Chapter 6 deals with the general
behaviour of the widely used second order type-I PLL. In this loop a LPF
is inserted between the PD and the VCO. The insertion of the LPF attenu-
atestherippleandalterstheloopdynamicsconsiderably.
Chapter 7 analyzes the synchronization of the second order type-I loop
at the third harmonic of the VCO center frequency. The analysis provides
the physical interpretation of the frequency acquisition procedure and
clarifies the effect of the initial condition and the LPF in the dynamic be-
haviouroftheloop.
Chapter 8 analyzes in depth the main synchronization of the second or-
der type-I loop. The analysis results in closed form expressions for the
hold-in, capture and tracking ranges, the lock-in time and the beat-note
frequency. In addition, the analysis of this loop provides the physical in-
terpretation of the fast (lock-in) and the slow (pull-in) capture procedure.
Furthermore, new criteria for the optimum design of the loop are sug-
gested.
Part IV. Second order type-II PLL: Chapter 9 treats the general be-
haviour of the second order type-II PLL. In this loop the LPF is replaced
by aPI-controller.Type-IIloopsareconsideredtobesuperiortothetype-I
loops, especially due to their enlarged capture and tracking ranges. How-
ever, the loop suffers from the saturation of the PI-controller, caused by
the ripple. To avoid saturation, the analog gain of the PI-controller should
be low. With a low analog gain the lock-in time increases considerably.
Preface IX
Thesaturationsplitsthehold-inrangeintwoparts:theunsaturatedandthe
saturated hold-in range. In the unsaturated hold-in the steady state phase
errorbetweenthereferenceandtheVCOsignalisequaleithertozeroorto
π/2. In contrast, in the saturated hold-in range this error becomes a func-
tionofthereferencefrequencyandtheloopbehavesasafirstorderone.
Chapter10analyzesthemainsynchronizationofthesecondordertype-
II PLL. This analysis results in closed form expressions for the hold-in
range, the tracking range and the lock-in time of the unsaturated loop. Fur-
thermore,thestabilitycriteriaofthelooparestated.
Chapter 11 analyzes briefly the synchronization of the second order
type-II loop at the third harmonic of the VCO center frequency. It is
proved that the harmonic hold-in range is large and depends strongly on
initialconditionsandontheanaloggainofthePI-controller.
Part V. Third order type-II PLL: Chapter 12 discusses the general
behaviour of the third order type-II PLL. In this loop a LPF is inserted be-
tween the PD and the PI-controller. The LPF reduces the ripple and the
saturationofthePI-controllerisprevented.
Chapter 13 treats the main synchronization of the third order type-II
PLL. The analysis results in closed formexpressions for the hold-in range,
the tracking range and the lock-in time of the unsaturated loop. Further-
more, the stability problem of this loop is detected and new criteria for the
loopdesignarestated.
Chapter14examinesbrieflythesynchronizationofthethirdordertype-
IIloopatthethirdharmonicoftheVCOcenterfrequency.Itisprovedthat
theharmonichold-inrangeissmallcomparedwiththemainhold-inrange.
The material contained in this book is the outcome of theoretical and
experimental research efforts that lasted for many years. I hope that it will
prove useful through suggesting new techniques in the analysis and syn-
thesisofphaselockedloops.
This book was first published in Greek by Tziolas Editions, Thessalo-
niki, Greece in the year 2000. The English translation was carried-out by
my sons Ioannis and Konstantinos Margaris, who I thank for their help. In
addition, I would like to thank my young friend Katerina-Galatea Xantho-
poulou for her linguistic advice and for improving the textual quality. Fi-
nally, I wish to thank my former students who contributed to this research
carried-outwithintheframeworkoftheirdiplomatheses.
Thessaloniki,December2003 NikolaosI.Margaris
Contents
Part I.Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 1
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1 PLL structure . . . . . . . . . . . . . . . . . . . . . . 4
1.2 PLLs classification . . . . . . . . . . . . . . . . . . . 5
1.3 The weakpoints of theclassicalPLL theory. . . . . . . 6
1.4 Scopeandmethodology . . . . . . . . . . . . . . . . . 7
References . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.PLLcomponents. . . . . . . . . . . . . . . . . . . . . . 13
2.1 PLL structure . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Voltagecontrolledoscillator. . . . . . . . . . . . . . . 13
2.2-1VCO representation. . . . . . . . . . . . . . . . 14
2.2-2 Transformationof theVCOequation . . . . . . . 16
2.2-3 AveragedVCOequations . . . . . . . . . . . . . 19
2.3 Phasedetector . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Low-pass filter. . . . . . . . . . . . . . . . . . . . . . 23
2.4-1Phaselead-lagfilter . . . . . . . . . . . . . . . . 23
2.4-2 PIcontroller . . . . . . . . . . . . . . . . . . . . 25
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 27
References . . . . . . . . . . . . . . . . . . . . . . . . . 28
Part ΙΙ.First order PLL. . . . . . . . . . . . . . . . . . . . . . . 31
3.Introduction tofirst order PLL. . . . . . . . . . . . . 33
3.1Representationof thefirst order loop.Generalform. . 33
3.2 Physicalinterpretationof thePLLmechanism. . . . . 36
3.2-1Main synchronization . . . . . . . . . . . . . . 36
3.2-2 Thirdharmonic synchronization. . . . . . . . . 38
3.3 Representationof thefirst order PLL.Specificform . 38
3.4Preliminary investigation . . . . . . . . . . . . . . . 40
XII Contents
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . 43
4.Main synchronization . . . . . . . . . . . . . . . . . . 45
4.1 Local stability . . . . . . . . . . . . . . . . . . . . . 45
4.1-1Equilibriumpoints . . . . . . . . . . . . . . . . 45
4.1-2 Effect ofparametersonequilibriumpoints . . . 47
4.1-3 Comment . . . . . . . . . . . . . . . . . . . . 51
4.1-4Local stabilityconditions . . . . . . . . . . . . 51
4.2 Hold-in range . . . . . . . . . . . . . . . . . . . . . 53
4.3 Trackingrange. . . . . . . . . . . . . . . . . . . . . 57
4.4Globaldynamicbehaviour . . . . . . . . . . . . . . . 63
4.4-1Transformationof theaveragedequations . . . . 63
4.4-2 Capturephenomenon . . . . . . . . . . . . . . 64
4.4-3 Detuning . . . . . . . . . . . . . . . . . . . . 72
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 74
References . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.Thirdharmonic synchronization . . . . . . . . . . . . 75
5.1 Local stability . . . . . . . . . . . . . . . . . . . . . 75
5.1-1Equilibriumpoints . . . . . . . . . . . . . . . . 76
5.1-2 Effect ofparameters onequilibriumpoints . . . 77
5.1-3 Local stabilityconditions . . . . . . . . . . . . 78
5.2 Hold-in range . . . . . . . . . . . . . . . . . . . . . 80
5.3 Trackingrange. . . . . . . . . . . . . . . . . . . . . 81
5.4Globaldynamicbehaviour . . . . . . . . . . . . . . . 83
5.4-1Transformationofaveragedequations. . . . . . 83
5.4-2 Capturephenomenon . . . . . . . . . . . . . . 83
5.4-3 Detuning. . . . . . . . . . . . . . . . . . . . . 84
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . 85
PartΙΙΙ.Secondorder type-IPLL. . . . . . . . . . . . . . . . . . 87
6. Introduction tosecondorder type-IPLL. . . . . . . . 89
6.1Representationof the secondorder type-IIPLL . . . . 90
6.1-1Exact equations . . . . . . . . . . . . . . . . . 91
6.1-2 Averagedequations . . . . . . . . . . . . . . . 92
6.2 Preliminaryinvestigation . . . . . . . . . . . . . . . 94
References. . . . . . . . . . . . . . . . . . . . . . . . . 99