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## Field and Network Theories in ECE

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Theories in ECE

Levent SEVGİ

1Electronics and Communication Engineering Dept, DOĞUŞ University, Zeamet Sokak, No 21, Acıbadem / KADIKOY - Istanbul

- 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.

- 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 =2c/ 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.

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

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

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

Incident

Transmitted

a1

b2

S22

S11

[S]

Port 2

Port 1

Reflected

Reflected

b1

a2

Incident

S12

Transmitted

Complexity Architecture

- Scattered (S) parameter model

Maxwell’s Equations and Decomposition

Full, Source-excited

Maxwell’s equations + BC

Longitudinal Fields

(dependent)

Transverse Source-excited

Maxwell’s 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)

- 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 z=ZL (left boundary) and z=ZR (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).

- 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.

- A generic problem for transmission lines is the Green’s function problem. Solving Sturm-Liouville equation when is a fixed parameter yields nothing but the Green’s function.
- The Green’s 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.
- Green’s function solution may also be built from eigenfunction solutions.
- Alternatively, eigenfunctions may also be derived from the Green’s 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.

- 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)

Example : Parallel plate waveguide

x

PEC

z

PEC

- Start with Maxwell’ equations

and obtain two sets of equations as

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, Maxwell’s (wave) equations.

Boundary conditions

+

wavenumber

Example : Parallel plate waveguide

- The Green’s function associated with this problem is:

+

Normalized Eigenfunciton

Z dependent coefficient

Example : Parallel plate waveguide

- Z dependent solution is

- and, finally the Green’s function is obtained as

- Ey may directly be obtained from the Green’s function as

Mode 5

Mode 25

f=300 MHz

Modal Caustics

Height [m]

a0 = -410-5

Normalized Mode Amplitude

Example #2: Open surface waveguide

15 km

Height [m]

100 NM 10-6 contribution

0 km

12 km

Normalized Field Strength

Software Calibration: SSPE vs. NM

- 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.

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