Slayt Basligi Yok

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Field and Network Theories in ECE
Levent SEVGI 1Electronics and Communication
Engineering Dept, DOGUS University, Zeamet Sokak,
No 21, Acibadem / KADIKOY - Istanbul
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Well-posed Problems

• A problem is said to be well-posed if the
  following conditions are met
• Existence implies that the conditions do not
  over determine the problem and the solution
  exists,
• Uniqueness implies that the conditions do not
  under determine the solution and it assures that
  the solution, no matter how obtained, is the
  correct and the only one.
• Stability implies that arbitrary small
  perturbations in data, e.g., sources, do not,
  for any physical quantity, let to infinity.
• Completeness, is to represent any excitation in
  terms of building blocks called Modes (i.e.,
  eigensolutions.

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Complexity Architecture

• Wave interaction with complex environmens poses
  major challenges to the analytic modeler.
  Complexity in the context of waves, encompasses
  many scales which are conveniently referenced
  to the relevant wavelengths ?2?c/? in the
  interrogating wave signal, where c is the
  wavespeed in a reference ambient medium and ?
  is the radian frequency.
• These relative scales si / ? can be associated
  with physical dimensions si ? di wavelengths si
  ? ?i in various materials temporal widths si ?
  Ti of the signal spectra sampling window widths
  si ? Wi in the processing data, etc.
• The wave modeler must decide how to
  parameterize a complex physical problem so as
  to take best advantage of the wave-based and
  computational tools at his disposal. It is
  natural to employ an architecture which
  decomposes the overall complex problem domain
  into simpler more tracktable interacting
  subdomain (SD) problems.

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Complexity Architecture

• A general framework for decomposing an overall
  complex problem space into interacting
  subdomains

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Guided waves and Reduction

• The steady-state EM vector fields excited by a
  specified electric and magnetic current
  distributions, J and M, respectively, are
  defined by Maxwells field equations

• on the perfectly conducting boundary of the
  uniform waveguide, the tangential component of
  the electric field must vanish

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Complexity Architecture

• Transmission line model

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Complexity Architecture

• Scattered (S) parameter model

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Maxwells Equations and Decomposition
Full, Source-excited Maxwells equations BC
Longitudinal Fields (dependent)
Transverse Source-excited Maxwells equations
BC
Eigensolutions (source-free)
Modal Amplitudes (scalar)
Source-free Transverse Field equations BC
TE Type (Dirichlet)
Transmission Line Equations
Transverse Vector Eigenfunctions (scalarization)
TM Type (Neumann)
Mixed (Cauchy)
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Guided waves and Reduction

• EIGENVALUE PROBLEM

(Dirichlet type)

• TRANSMISSION LINE PROBLEM

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Sturm-Liouville Equation (SL)

• Each 1D problem in a coordinate-separable
  reduced 2D or 3D problem is parameterized by SL
  theory which deals with eigenspectra of 1D
  linear second-order diferential operators in
  bounded and unbounded domains.

with suitable boundary condititons at zZL (left
boundary) and zZR (right boundary) represents
our EM equations. Depending on the p(z), q(z) and
?(z) this may be Helmholtz wave equation, Laplace
equation, diffusion equation, etc.

• This represents a Transmission Line problem if
  ? is a fixed parameter.

• It is an Eigenvalue problem if ? is a free
  parameter that does not equal to an eigenvalue
  (Since eigenvalues are real, choosing Im ??0
  guarantees uniqueness).

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Alternative representations

• Phase space wave dynamics can be organized
  around two complementary phenomenologies -
  progressing and oscillatory - which are closely
  related to local vs. global descriptions.
• Progressing wave objects (rays or wavefronts)
  are responsible for point-to-point propagation
  and sample the physical environmentlocally
  along their trajectories.
• Oscillatory wave objects (modes or resonances)
  form standing waves over extended (global)
  portions of the physical environment.
• An alternative approach to the local-global
  parameterization of wave phenomena is through
  the use of Poisson summation, which converts an
  infinite or truncated n-sum (poorly convergent)
  series into an m-sum (good convergent ) series
  in Fourier domain.

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Sturm-Liouville Equation

• A generic problem for transmission lines is the
  Greens function problem. Solving
  Sturm-Liouville equation when ? is a fixed
  parameter yields nothing but the Greens
  function.
• The Greens function is the response of a
  linear system to a point source of unit
  strength. A source of unit strength at a
  position a along, for example, z coordinate
  can be represented by the Dirac delta function.
• Greens function solution may also be built
  from eigenfunction solutions.
• Alternatively, eigenfunctions may also be
  derived from the Greens function solution.
• This illustrates that a guided wave problem may
  be handled as either an eigenvalue problem or
  transmission line problem, since the former
  also yields the latter and vice versa.

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Normal Mode Solution

• NM is the solution of source-free wave equation
  in a coordinate system that yields separation of
  variables
• NM satisfy transverse boundary conditions on
  each transverse cross-section
• NM confinement may be due to fixed transverse
  boundaries and/or mode dependent virtual
  boundaries (refractive confinement)
• NM propagate in longitudinal direction with
  distinct propagation constants
• Each NM carries finite energy (can be
  normalized) and is independent of every other
  NM (orthogonality)

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Example Parallel plate waveguide
x
PEC
z
PEC

• Start with Maxwell equations

and obtain two sets of equations as
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Example Parallel plate waveguide

• Set 1 TMz

Set 1 and Set 2 are decoupled

• Set 2 TEz

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Example Parallel plate waveguide

• Any of Jx, My or Jz excite SET 1 (TMz)
• Any of Jy, Mx or Mz excite SET 2 (TMz)
• Lets look at TEz with Jy source only and start
  with second order decoupled, Maxwells (wave)
  equations.

Boundary conditions

wavenumber
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Example Parallel plate waveguide

• The Greens function associated with this
  problem is

Normalized Eigenfunciton
Z dependent coefficient
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Example Parallel plate waveguide

• Using the identities

one obtains

• Defining and yields

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Example Parallel plate waveguide

• Z dependent solution is

• and, finally the Greens function is obtained as

• Ey may directly be obtained from the Greens
  function as

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Example Parallel plate waveguide

• For a line source at (x,z) the solution will
  then be

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Example 2 Open surface waveguide
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Software Calibration SSPE vs. NM
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Conclusions and Discussions

• Field and network theories are two fundamental
  approaches in ECE.
• Any ECE problem can be postulated via one of
  these two approaches, the solution can be
  derived, and may then be transformed into the
  other.
• The problem at hand, the parameter regime, or
  the geometry give clues to choose the best
  suitable approach.