Advance Elements of Optoisolation Circuits: Nonlinearity Applications in Engineering

837 Pages·2017·15.97 MB·English

by  Ofer Aluf (auth.)

Ofer Aluf
Advance
Elements of
Optoisolation
Circuits
Nonlinearity Applications in Engineering
Advance Elements of Optoisolation Circuits
Ofer Aluf
Advance Elements
of Optoisolation Circuits
Nonlinearity Applications in Engineering
123
Ofer Aluf
Netanya
Israel
ISBN978-3-319-55314-6 ISBN978-3-319-55316-0 (eBook)
DOI 10.1007/978-3-319-55316-0
LibraryofCongressControlNumber:2017934211
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Preface
This book on advance optoisolation circuits and nonlinearity applications in engi-
neering covers and deals two separate engineering and scientific areas and what
happens in between. It is a continuation of the first book “Optoisolation Circuits
Nonlinearity Applications in Engineering”. It gives advance analysis methods for
optoisolation circuits which represent many applications in engineering.
Optoisolation circuits come in many topological structures and represent many
specificimplementationswhichstandthetargetengineeringfeatures.Optoisolation
circuits includes phototransistor and light-emitting diode (optocouplers), both
coupledtogether.Therearemanyothersemiconductorswhichincludeoptoisolation
topologies like photo SCR, SSR, photodiode, etc. The basic optocoupler can be
characterizedbyEbers–Mollmodelandtheassociatedequations.Theoptoisolation
circuits include optocouplers and peripheral elements (capacitors, inductors, resis-
tors, switches, operational amplifiers, etc.). The optoisolation circuits analyze as
linearandnonlineardynamicalsystemsandtheirlimitcycles,bifurcation,andlimit
cycle stability analysis by using Floquet theory. This book is aimed at newcomers
to linear and nonlinear dynamics and chaos advance optoisolation circuits. Many
advanceoptoisolationcircuitsexhibitlimitcyclebehavior.Alimitcycleisaclosed
trajectory (system phase space V (t), V (t) voltages in time are coordinates); this
1 2
meansthatitsneighboringtrajectoriesarenotclosed—theyspiraleithertowardsor
awayfromthelimitcycles.Thus,limitcyclescanonlyoccurinthoseoptoisolation
circuits which exhibit nonlinearity (nonlinear systems). Optoisolation circuits
exhibit many kinds of bifurcation behaviors. The basic definition of bifurcation
describes the qualitative alterations that occur in the orbit structure of a dynamical
system as the parameters on which the optoisolation system depends are varied.
There are many bifurcations we discuss related to optoisolation systems:
cusp-catastrophe, Bautin bifurcation, Andronov–Hopf bifurcation, Bogdanov–
Takens (BT) bifurcation, fold Hopf bifurcation, Hopf–Hopf bifurcation, Torus
bifurcation (Neimark–Sacker bifurcation), and saddle–loop or homoclinic bifurca-
tion. Floquet theory helps us to analyze advance optoisolation systems. Floquet
theory is the study of the stability of linear periodic systems in continuous time.
Another way to describe Floquet theory: it is the study of linear systems of
v
vi Preface
differential equations with periodic coefficients. Floquet theory can be used for
anything foe which you would use linear stability analysis, when dealing with
optoisolation periodic systems. Many optoisolation systems are periodic and con-
tinue in time; therefore Floquet theory is an ideal way for behavior and stability
analysis. There are many optoisolation systems which contain periodic forcing
source and can be analyzed by this theory. Floquet theory is also implemented in
OptoNDRcircuitanalysis.Optoisolationcircuitswithperiodiclimitcyclesolutions
orbital stability is part of advances system. Optoisolation circuits bifurcation cul-
minatingwithperiodiclimitcycleoscillation.Theroutebywhichchaosarisesfrom
mixed-mode periodic states in optoisolation systems. The optoisolation system
displays a rich variety of dynamical behavior including simple oscillations,
quasiperiodicity, bi-stability between periodic states, complex periodic oscillations
(including the mixed-mode type), and chaos. The route to chaos in this optoiso-
lation system involves a torus attractor which becomes destabilized and breaks up
into a fractal object, a strange attractor. Optoisolation systems exhibit sequence
leading from a mixed-mode periodic state to a chaotic one in the staircase region
and find a familiar cascade of periodic-doubling bifurcations, which finally cul-
minate in chaos. Optoisolation circuits with forced van der Pol sources exhibit a
variety of implementations in many engineering areas. Optoisolation circuit aver-
aging analysis and perturbation from geometric viewpoint.
In this book we try to provide the reader with explicit advance procedures for
application of general optoisolation circuits mathematical representations to par-
ticular advance research problems. Special attention is given to numerical and
analytical implementation of the developed techniques.
Let us briefly characterize the content of each chapter.
Chapter 1. Optoisolation Circuits with Limit Cycles. In this chapter
optoisolation circuits with limit cycles aredescribed andanalyzed.Alimitcycleis
an closed trajectory (system phase space V (t), V (t) voltages in time are coordi-
1 2
nates);thismeansthatitsneighboringtrajectoriesarenotclosed—theyspiraleither
towards or away from the limit cycle. Thus, limit cycles can only occur in those
optoisolation circuits which exhibit nonlinearity (nonlinear systems). In contrast a
linear system exhibiting oscillations closed trajectories are neighbored by other
closedtrajectories(Example:dh/dt=f(h)).Astablelimitcycleisonewhichattracts
all neighboring trajectories. A system with a stable limit cycle can exhibit
self-sustainedoscillations.Neighboringtrajectoriesarerepelledfromunstablelimit
cycles.Half-stablelimitcyclesareoneswhichattracttrajectoriesfromtheoneside
andrepelthoseontheother.Ifallneighboringtrajectoriesapproachthelimitcycle,
the limit cycle is stable or attracting. Otherwise the limit cycle is unstable, or in
exceptional cases, half stable. An optoisolation circuit which exhibits stable limit
cycles oscillate even in the absence of external periodic forcing (wave generator
f(t)). Since limit cycles are nonlinear phenomena; they cannot occur in linear sys-
tems.Alimitcycleoscillationsaredeterminedbythestructureofthesystemitself.
Weuseinnonlineardynamicssomeacronyms:Stablelimitcycle(SLC),half-stable
limit cycle (HLC), unstable limit cycle (ULC), unstable equilibrium point (UEP),
stable equilibrium point (SEP), half-stable equilibrium point (HEP).
Preface vii
Chapter2.OptoisolationCircuitsBifurcationAnalysis(I).Inthischapterwe
discuss various bifurcations which exhibit by optoisolation circuits. The first is
cusp-catastrophe which occurs in a one-dimensional state space (n = 1) and
two-dimensionalparameterspace(p=2).IthasanequilibriummanifoldinR2xR:
dV¼fðV;C ;C Þ V 2Rn¼1; f C ;C g2Rp¼2; M ¼fðC ;C ;VÞjC þC (cid:2)V (cid:3)
dt 1 2 1 2 1 2 1 2
V3 ¼0g, where C and C are two control parameters and V is the system state
1 2
variable. A two-parameter system near a triple equilibrium point known as cusp
bifurcation (equilibrium structure). We can consider system with a higher dimen-
sional state space and parameter space. The Bautin bifurcation is equilibrium in a
two-parameter family of system’s autonomous ODEs at which the critical equi-
librium has a pair of purely imaginary eigenvalues and the first Lyapunov coeffi-
cientfortheAndronov–Hopfbifurcationvanishes.Thisphenomenonisalsocalled
the generalized Hopf (GH) bifurcation. Bogdanov–Takens (BT) bifurcation is a
bifurcationofanequilibriumpointinatwo-parameterfamilyofautonomousODEs,
critical equilibrium has a zero eigenvalue of multiplicity two.
Chapter 3. Optoisolation Circuits Bifurcation Analysis (II). In this chapter
we discuss various bifurcations which exhibit by optoisolation circuits. The fold
Hopfbifurcationisabifurcationofanequilibriumpointinatwo-parameterfamily
ofautonomous ODEsatwhichthecritical equilibriumhasazeroeigenvalueanda
pair of purely imaginary eigenvalues. The Hopf–Hopf bifurcation is when the
critical equilibrium has two pairs of purely imaginary eigenvalues in system’s
two-parameter family of autonomous differential equations. Torus bifurcations
(Neimark–Sacker bifurcations) of the limit cycles generated by the Hopf bifurca-
tions. This curve of torus bifurcations is transversal to the saddle–node and
Andronov–Hopf bifurcation curves. The torus bifurcation generates an invariant
two-dimensional torus. The invariant torus disappears via either a “heteroclinic
destruction”ora“blow-up”.Saddle–looporhomoclinicbifurcationiswhenpartof
a limit cycle moves closer and closer to a saddle point. At the homoclinic bifur-
cation the cycle touches the saddle point and becomes a homoclinic orbit (infinite
period bifurcation).
Chapter 4. Optoisolation Circuits Analysis Floquet Theory. In this chapter
optoisolation circuits are analyzed by using Floquet theory. Floquet theory is the
study of the stability of linear periodic systems in continuous time. Floquet
exponents/multipliers are analogous to the eigenvalues of Jacobian matrices of
equilibriumpoints.AnotherwaytodescribeFloquettheory:itisthestudyoflinear
systems of differential equations with periodic coefficients. Floquet theory can be
used for anything for which you would use linear stability analysis, when dealing
with a periodic system. Although Floquet theory is a linear theory, nonlinear
models can be linearized near limit cycle solutions to enable the use of Floquet
theory. Floquet theory deals with continuous-time systems. The theory of periodic
discrete-time systems is closely analogous. In that case, one can multiply the T
transitionmatricestogethertodeterminehowaperturbationchangesoveraperiod,
whichissimilartofindingthefundamentalmatrix.OnelimitationofFloquettheory
is that it applies only to periodic systems. Although many systems experience
viii Preface
periodic forcing, others experience stochastic or chaotic forcing. In these cases,
more general Lyapunov exponents are described which play the role of Floquet
exponents. Conceptually similar to Floquet exponents, Lyapunov exponents are
more challenging to compute numerically because, instead of calculating how a
perturbation grows or shrinks over one period, this must be done in the limit at
T!∞.
Chapter 5. Optoisolation NDR Circuits Behavior Investigation by Using
Floquet Theory. In this chapter we discuss optoisolation NDR circuits behavior
analysis by using Floquet theory. Floquet theory is the study of the stability of
linear periodic systems in continuous time. Floquet exponents/multipliers are
analogous to the eigenvalues of Jacobian matrices of equilibrium points. We con-
sider an OptoNDR circuit with storage elements (variable capacitor and variable
inductance) in the output port. OptoNDR circuits’ nonlinear models is linearized
near limit cycle solutions to enable the use of Floquet theory. We analyze Chua’s
circuit and find fixed points and stability. We replace Chua’s diode by OptoNDR
element in Chua’s circuit. The circuit contains three energy storage elements and
exhibits different and unique variety of bifurcations and chaos among the
chaos-producing mechanism with OptoNDR element. We consider an OptoNDR
circuit with two storage elements and find behavior, fixed points, and stability.
OptoNDR circuit’s two variables are simulated and analyzed by using Floquet
theory. We use Floquet theory to test the stability of optoisolation circuits limit
cycle solution.
Chapter 6. Optoisolation Circuits with Periodic Limit-Cycle Solutions
Orbital Stability. In this chapter we implement systems with periodic limit cycle
by using optoisolation circuits. We discuss periodic and solutions orbital stability.
Periodicorbitsarenonequilibriumtrajectories X(t)thatsatisfyX(T)=X(0)forsome
T > 0. The basic behavior of planar cubic vector field and van der Pol equation is
analyzed and discussed for limit cycle and stability. Van der Pol system is
implementedbyusingOptoNDRelementandthelimitcyclesolutionisdiscussed.
Glycolytic oscillation is the repetitive fluctuation of in the concentrations of
metabolites. The modeling glycolytic oscillation is presented and discussed in
controltheoryanddynamicalsystems.Themodelexplainssustainedoscillationsin
the yeast glycolytic system. We discuss the differential equations model of gly-
colytic oscillator and find fixed points, stability, and limit cycle. We present and
discuss optoisolation glycolytic circuits limit cycle solution.
Chapter7.OptoisolationCircuitsPoincareMapsandPeriodicOrbit.Inthis
chapter we deal with optoisolation circuits and investigate periodic orbit and
Poincare map. A Poincare map is a discrete dynamical system with a state space
that is one dimension smaller than the original continuous dynamical system. We
use it for analyzing the original system. We use Poincare maps to study the flow
nearaperiodicorbitandtheflowinsomechaoticsystem.Systemphysicalbehavior
in time is described by a dynamical system. A dynamical system is a function of
/ðt;XÞ,definedforallt 2RandX 2E 2Rn.APoincaremapisaveryusefultool
to study the stability and bifurcations of system periodic orbit. Optoisolation van
derPoloscillatorcircuitwithparallelcapacitorC dynamicalbehaviorisinspected
2
Preface ix
byusingaPoincaremap.Liautonomoussystemwithtoroidalchaoticattractorsisa
typicaldynamicalsystem.TheLisystemglobalPoincaresurfaceofsectionhastwo
disjoint components. A similarity transformation in the phase space emphasizes
symmetry of the attractor. One way for the production of oscillations in L-C net-
worksistoovercomecircuitlossesthroughtheuseofdesigned-inpositivefeedback
or generation. Negative resistance oscillator with OptoNDR device has better
performance characteristics cancellation in terms of resistive losses in oscillators
than oscillators that are not based on NDR devices. We investigate it by using
Poincare map and find periodic orbit.
Chapter 8. Optoisolation Circuits Averaging Analysis and Perturbation
from Geometric Viewpoint. In this chapter we discuss optoisolation circuits
averaging analysis and perturbation from geometric viewpoint. In many perturbed
systems, we start with a system which includes known solutions and add small
perturbationstoit.Thesolutionsofunperturbedandperturbedsystemsaredifferent
and system with small perturbation has different structure of solutions. In pertur-
bation theory we use approximate solution by discussing of the exact solution. An
approximation of the full solution A, a series in the small parameters (e), like the
P1
following solution A¼ A (cid:2)ek. A is the known solution to the exactly soluble
k 0
k¼0
initial problemandA ;A ;...represent thehigher order terms. Higherorder terms
1 2
becomesuccessivelysmallerforsmalle.Ineverydynamicalsystemweneedtouse
theglobalexistencetheorems.Thesystemsolutionisalteredfornon-zerobutsmall
e. We analyze OptoNDR circuit van der Pol by using perturbation method and we
usemultiplescaleanalysisforconstructionuniformorglobalapproximatesolutions
forbothsmallandlargevaluesofindependentvariablesofOptoNDRcircuitvander
Pol.OptoNDRcircuitforcedvanderPolisanalyzedbyusingperturbationmethod
time scale.Weconsiderforced vanderPolsystemwith forcing termF(cid:2)cos(x(cid:2)t).If
“F”issmall(weakexcitation),itseffectdependsonwhetherornotxisclosetothe
natural frequency. If it is, then the oscillation might be generated which is a per-
turbation of the limit cycle. If “F” is not small (hard excitation), then the “natural
oscillation” mightbe extinguished.
Chapter 9. Optoisolation Advance Circuits—Investigation, Comparison
and Conclusions. In this chapter, we summarized the main topics regarding
optoisolation advance circuits; inspect behavior, dynamics, stability, comparison,
and conclusions. Optoisolation advance circuits are an integral part of every
industrial system. An optoisolation circuit can have limit cycles which we can
analyze. Additionally there are many bifurcations that can characterize optoisola-
tion circuits. Floquet theory is analyzed in many systems which include optoiso-
lationcircuits.OptoisolationNDRcircuitsbehaviorisinvestigatedbyusingFloquet
theory and periodic limit cycle solutions orbital stability is discussed. We present
optoisolation circuits by Poincaremaps and periodic orbit. Averaging analysisand
perturbation from geometric viewpoint are implemented in our circuits.
Netanya, Israel Ofer Aluf
Contents
1 Optoisolation Circuits with Limit Cycles.. .... .... .... ..... .... 1
1.1 Optoisolation Circuits with Limit Cycles ... .... .... ..... .... 1
1.2 Optoisolation Circuits with Limit Cycles Stability Analysis.. .... 29
1.3 Poincare–Bendixson Stability and Limit Cycle Analysis .... .... 34
1.4 Optoisolation Circuits Poincare–Bendixson Analysis .. ..... .... 51
1.5 Optoisolation Nonlinear Oscillations Lienard Circuits . ..... .... 70
1.6 Optoisolation Circuits with Weakly Nonlinear Oscillations .. .... 83
1.7 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 90
2 Optoisolation Circuits Bifurcation Analysis (I). .... .... ..... .... 93
2.1 Cusp Bifurcation Analysis System .... .... .... .... ..... .... 93
2.2 Optoisolation Circuits Cusp Bifurcation Analysis. .... ..... .... 98
2.3 Bautin Bifurcation Analysis System... .... .... .... ..... .... 114
2.4 Optoisolation Circuits Bautin Bifurcation Analysis.... ..... .... 126
2.5 Bogdanov–Takens (Double-Zero) Bifurcation System . ..... .... 140
2.6 Optoisolation Circuits Bogdanov–Takens (Double-Zero)
Bifurcation . .... .... ..... .... .... .... .... .... ..... .... 159
2.7 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 169
3 Optoisolation Circuits Bifurcation Analysis (II) .... .... ..... .... 175
3.1 Fold-Hopf Bifurcation System ... .... .... .... .... ..... .... 175
3.2 Optoisolation Circuits Fold-Hopf Bifurcation.... .... ..... .... 185
3.3 Hopf–Hopf Bifurcation System... .... .... .... .... ..... .... 197
3.4 Optoisolation Circuits Hopf–Hopf Bifurcation System. ..... .... 206
3.5 Neimark–Sacker (Torus) Bifurcation System .... .... ..... .... 227
3.6 Optoisolation Circuits Neimark–Sacker (Torus) Bifurcation.. .... 254
3.7 Exercises .. .... .... ..... .... .... .... .... .... ..... .... 275
4 Optoisolation Circuits Analysis Floquet Theory .... .... ..... .... 281
4.1 Floquet Theory Basic Assumptions and Definitions... ..... .... 281
4.2 Optoisolation Circuit’s Two Variables with Periodic Sources. .... 289
xi