Chaos Control in a Transmission Line Model

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Chaos Control in a Transmission Line Model
Dr. Ioana Triandaf Nonlinear Dynamical Systems
Section Code 6792 Naval Research
Laboratory Washington, DC 20375 Ioana.triandaf_at_nr
l.navy.mil IEEE International Conference on
Electronics,Circuits and Systems, December 12-15,
2010, Athens, Greece Work supported by Office
of Naval Research
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Problem and Objective

• Problem Modeling electromagnetic interference
• Analytical and numerical models exist only for
  simple networks of electronic devices.
• Prominence was given to computational methods
  rather than to the analysis of the
• qualitative behavior of the solutions.
• Many open questions remain in the area of
  relating field tests to current theory for the
• analysis, design, and control of dynamically
  interacting nonlinear networks.
• Understanding failure mechanisms in such
  networks is highly relevant to defense as well
• as commercial applications.

• Objective Predict disruption and damage in
  electronic devices
• Obtain a qualitative analysis of solutions of
  circuit networks over a wide parameter space.
• Understanding at a fundamental level of how the
  transition to damage occurs
• Our intention is to understand the underlying
  dynamics, forecast future effects, and control
  effects.

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Talk Outline

• Statement of the problem and motivation
• Present the broad research goal
• Transmission line equations and properties
• Chaotic dynamics in an infinite-dimensional
  electromagnetic system
• Control of the chaotic behavior

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The Dynamic Equations of Transmission Lines
Approach assume that the transverse electric
and magnetic fields surrounding the conductor is
transverse to the conductor (the quasi-TEM
approximation), these are dominant modes when the
cross-sectional dimensions of the guiding
structure are less than the smallest
characteristic wavelength of the electromagnetic
field propagating along it. Ideal transmission
lines ideal guiding structures, model
interconnections without losses, uniform in space
and with parameters independent of frequency. The
equations for the voltage and current
distribution are where L is the
inductance per-unit-length and C is the
capacitance per-unit-length.
Sketch of a two-conductor transmission line
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Lossy Transmission Lines with Distributed
Sources

• Lossy transmission lines
• R the per-unit-length resistance and G the
  per-unit-length conductance
• Transmission lines with distributed sources
• The distributed sources depend on the incident
  electromagnetic field and on the structure
• of the guiding system modeled by the transmission
  line.

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DAlembert Solution of Two-Conductor
Transmission Lines
Equations

• The general solution of the line equation
  in dAlembert form is
• is the forward
  voltage wave, is the
  backward voltage wave
• This solution can be represented in terms of
  forward and backward voltage waves only,
• we can also represent it through the
  superposition of forward and backward current
  waves
• and are univocally determined by
  the initial conditions

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The Case of a Semi-infinite Line Connected to a
Nonlinear resistor

• We consider the case in which
  the semi-infinite line is connected to a
  nonlinear resistor
• Impose that the solution satisfies
  , g is the
  characteristic of the resistor. The equation for
  is
• 
  
• There may be multiple solution depending on
  the characteristic curve of the resistor.

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A Semi-infinite Line Connected to a Nonlinear
Resistor in Parallel to a Capacitor

• We consider the case in which the
  semi-infinite line is connected to a nonlinear
  resistor connected in parallel to a linear
  capacitor. The equivalent circuit is illustrated
  as
• The equation for is
• 
  
• this equation must be solved together with
  the initial condition for .
• Because of the presence of the capacitor the
  relation between and is no longer of
  the
• instantaneous type, but is of the functional
  type, with memory.

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Chaotic Dynamics in an Infinite-Dimensional
Electromagnetic System

• Bifurcation and chaos phenomena theoretically
  observed in a simple electromagnetic
• system consisting of a linear resistor and a
  pn-junction diode connected by a transmission
• line
• The system is infinite-dimensional because of the
  presence of the transmission line and the
• nonlinearity arises due to the pn-junction diode.
• We solve the above equation obeying initial
  conditions ,
• and nonlinear boundary conditions
• L. Corti, L. De Menna, G. Miano and L. Verolino,
  IEEE Trans on Circuits and Systems, Vol. 41, No.
  11, November 1994
• 
  

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Solving the Transmission Line Initial Boundary
Value Problem

• By imposing initial conditions and to
  the general solution for
• we obtain for
• By imposing the voltages and currents at line
  ends, , and , we obtain the
  relation
• between the state of the line and the electrical
  variables at the line ends, for in
  implicit form
• 
  
• The relationship between the value of the
  backward wave at a boundary point and the value
  of this wave at a point
• inside the spatial domain is given by

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Formulation of the Problem as a Dynamical System

• The general solution of the Telegraphers
  equation is
• By imposing boundary conditions
• 
  
• we obtain the nonlinear implicit functional
  equation
• We are making it explicit in the form

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Formulation of the Problem as a Dynamical System

• The equation
  
• provides a convenient way to compute the forward
  travelling wave at the point
• and at time when its value at time
  is known.
• We formulate the functional equation as a
  recurrence relation in which the time is
• discretized obtaining
• The dynamic of the voltage at and at
  is given by
  

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The pn-junction Diode Dynamics

• We investigate the system with a pn-junction
  diode defined by the constitutive equation
  
• where is the saturation current and the
  thermal voltage. The maximum of the function
• F is reached when the incremental resistance of
  the diode is equal to the characteristic
• impedance.
• In this case the Poincare map
• becomes

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Chaotic Dynamics of the System

• If the linear resistor is active ,
  oscillations and chaotic motion appear.
• We consider corresponding to
  As increases, a period-doubling
  cascade
• of bifurcations forms, leading to chaos.
• The solution a
• 
  

Spatial profile of the wave at
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Control of the Chaotic Dynamics of the System

• We stabilize a period one orbit of the map
  , using small
  fluctuations of the parameter for
• each spatial point.
• 
  
• Tracking unstable orbits in
  experiments by Schwartz Ira B. and Triandaf
  Ioana , Phys. Rev. A, vol. 46, number 12,
  (7439-7444) , Dec 199.
• Tracking sustained chaos A
  segmentation method, by Triandaf, Ioana and
  Schwartz, Ira B., Phys. Rev. E, vol. 62, number
  3, (3529-3534), Sep 2000.
• Tracking unstable steady states
  Extending the stability regime of a multimode
  laser system, by Gills, Zelda and Iwata,
  Christina and Roy, Rajarshi
• and Schwartz, Ira B. and
  Triandaf, Ioana , Phys. Rev. Lett., vol. 69,
  number 22, (3169-3172), 1992.
• Quantitative and qualitative
  characterization of zigzag spatiotemporal chaos
  in a system of amplitude equations for nematic
  electroconvection

Controlled spatial profile of the wave
Spatial profile of the wave at
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Control of the Chaotic Dynamics of the System

• We stabilize a period one orbit of the map
  , using small
  fluctuations of the parameter for
• each spatial point.
• The relationship between the value of the forward
  wave at a boundary point and the value of this
  wave at a point inside
• the spatial domain is given by
• 
  

Spatial profile of the forward at
Controlled spatial profile of the wave
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The Control Method of the Chaotic Dynamics

• We stabilize a period one orbit of the map
  , using small
  fluctuations of the parameter for
• each spatial point.
• The fluctuation in for a given spatial
  point is given by
  , is the fixed point of the
  map,
• is the unstable eigenvalue of the
  fixed point,
• measures the local drift in the fixed
  point.
• 
  

Control parameters over the spatial domain
The maps used to determine
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Control of the Chaotic Dynamics of the System

• We stabilize a period one orbit of the map
  , using small
  fluctuations of the parameter for
• each spatial point.
• The relationship between the value of the forward
  wave at a boundary point and the value of this
  wave at a point inside
• the spatial domain is given by
• 
  

Spatial profile of the wave at
Controlled spatial profile of the wave
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Control of the Chaotic Dynamics of the System

• We stabilize a period one orbit of the map
  , using small
  fluctuations of the parameter for
• each spatial point.
• The relationship between the value of the forward
  wave at a boundary point and the value of this
  wave at a point inside
• the spatial domain is given by
• 
  

Spatial profile of the wave at
Controlled spatial profile of the wave
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Summary of our Approach

• We considered simple electromagnetic networks
  modelled by the wave equation with nonlinear
  boundary conditions.
• Objectives
• Recast the equations as lower dimensional
  systems, possibly maps.
• Study chaotic behavior of networks.
• Design algorithms that mimick disruption of
  networks in real devices.
• Understand how to solve coupled problems of a
  profoundly different nature
• Transmission line equations are linear and
  time-invariant pdes of hyperbolic type
• Lumped circuits equations are algebraic odes,
  time-varying and nonlinear
• Derive representations that take into account
  only the terminal behaviour
• of the transmission line
• Describe terminal behaviour by linear
  algebraic difference equations with one delay
• Study the existence and uniqueness of the
  difference-delay equations or solve in the
• multi-valued case
• Explain the occurrence of chaos
  encountered typically when frequencies increase

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Conclusions

• We have presented a chaos control method applied
  to a simple electromagnetic system.
• Control is achieved at times which are integer
  multiples of the round-trip time of the wave
  along the transmission line.
• The current method will be extended to the full
  simulation and will provide valuable insight on
  how to achieve control in that case.
• The understanding gained will be used in models
  derived from basic principles and a full
  stability analysis of solutions will be
  performed, leading to a deterministic approach to
  predict upset and failure of electronic networks.
• Gain understanding of disruption in circuits at
  a fundamental level possibly avoiding intensive
  computing and data storage required by
  probabilistic techniques.
• Prediction of conditions that lead to upset in
  electronic device, prediction of pathways to
  failure in network of circuits.

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Radiating Disturbances

• Examples of radiated disturbances
• crosstalk between circuits, problems related to
  printed circuit board, surges produced by
  switching operations, electrical short produced
  by a conductor fault in a system.
• antenna radiation, lightning or nuclear effects
• interconnection between two computer boards
• radiation of radio emitters, mobile radio
  communication, radar interference

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Multiconductor Transmission Lines Equations

• Transmission line having (n1) conductors
• The voltage and the current are
  vectors,
• , the per-unit-length
  self-inductance and
• , is the per-unit-length
  self-capacitance.

The equations for multiconductor transmission
lines
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Two-conductor Transmission Lines as Two - Ports

• Transmission line connecting generic lumped
  circuits
• The general solution of the line equations is
• where .
• The voltage and current distributions along the
  line are completely identified by the
• functions and and
  viceversa. We consider them as state variables of
  the
• line.