ieee ap-ursi ohio meeting sevgi, akleman, felsen field and network theories in ece theories in ece...

IEEE AP-URSI Ohio Meeting Sevgi, Akleman, Felsen

Field and NetworkField and Network

Theories in ECETheories 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.

Well-posed Problems


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.

Complexity Architecture



R1

R2

R3

R0

B12

B13

B23B21

B31

B32

B02

B20

B10

B01

B03

B30

Partitioning the Problem Space Ri : Subdomain, I=1,2 .. Bij : Interface between subdomains R i and Rj

Connection Relations(for fields)

Complex domainSegmentation

Subdomain equations(matrix form)

Networktopology

Subdomain Relations(for fields)

NetworkSolution

Field equation inTableau form

Discretization

Options:

Topological alternativesNumerical alternatives

Accesible andinternal observables:

Primary and secondaryfield variables

Alternative Green’sfunctions

Choice of field basisfunctions

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:

Guided waves and Reduction

)()()( rMrHjrE

)()()( rJrEjrH

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

S on 0 )(rEn

tjerEtrE ),(),(

tjerHtrH ),(),(



Transmission line model

L

Z0, = + j

Source

Load

z

ZL

x



Scattered (S) parameter model

[S]Port 1 Port 2

S12

S21

S22S11

Transmitted

TransmittedIncident

Incident

a1

a2

b1

b2

Reflected Reflected


Full, Source-excited Maxwell’s equations + BC

Longitudinal Fields(dependent)

Modal Amplitudes(scalar)

Transmission Line Equations

Source-free Transverse Field equations + BC

Transverse Vector Eigenfunctions(scalarization)

TE Type (Dirichlet)

TM Type (Neumann)

Transverse Source-excited Maxwell’s equations + BC

Eigensolutions (source-free)

Mixed (Cauchy)

Maxwell’s Equations and Decomposition


EIGENVALUE PROBLEM

Guided waves and Reduction

TRANSMISSION LINE PROBLEM

iiiii vIZj

dz

dV

iiiii iVYjdz

dI

S inside 022 ),(' yxk itit

s on 0 ),( yxi (Dirichlet type)


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.

)'();',()()()( zzzzgzzqdz

dzp

dz

d

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 problemTransmission Line problem if is a fixed parameter.

It is an EigenvalueEigenvalue problem problem if is a free parameter that does not equal to an eigenvalue (Since eigenvalues are real, choosing Im {}0

guarantees uniqueness).


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.


Sturm-Liouville Equation

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.


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)


Start with Maxwell’ equations

Example : Parallel plate waveguide

z

x

PEC

PEC

0y

MHjE

JEjH

and obtain two sets of equations as


Set 1: TMz


Set 2: TEz

xxy

EjJz

H0

zzy

EjJx

H0

xxy

HjMz

E0

yyzx HjMx

E

z

E0

yyzx EjJx

H

z

H0

zzy

HjMx

E0

Set 1 and Set 2 are decoupled


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.


yy JjEkzx

02

2

2

2

2

x

yx M

z

E

jH

0

1

z

yz M

x

E

jH

0

1

+ axEy ,00 at

Boundary conditions

0022 k

wavenumber

0

yz

ELim


The Green’s function associated with this problem is:


)'()'()',';,( zzxxzxzxgkzx

22

2

2

2axg ,00 at +

0

)',';,( zxzxgLimz

a

xnSin

a

(x)U nn

Normalized Eigenfunciton

Z dependent coefficient

)z',x'(z;(x)Un

n

nzxzxg )',';,(

(x)dx))Uz',x'z;g(x,)z',x'(z; na

n0

0n if

0n if

1

2n


Using the identities


)z',x'(z;na

nzxzxg

x

2

2

2 )',';,(

one obtains

)'()'()',';( zzxUzxza

nk

dz

dnn

2

22

2

Defining and yields 2

22

a

nk

)'(xUn

nn

)'()';( zzzzdz

dn

22

2

)'()()'( xUxUxx nnn

1


Z dependent solution is


02

Im)';(||

, j

ezz

zzj

n

and, finally the Green’s function is obtained as

a

x'nSin

a

xnSin

n

zzjn

j

e

azxzxg

2

||)',';,(

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

dAzxzxgzxJjEAyy )',';,()','(0


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


a

x'nSin

a

xnSin

n

zzj

y j

e

ajE

2

20

||

z

x

PEC

PEC

0y


Mode 1 Mode 5 Mode 25H

eigh

t [m

]

Normalized Mode Amplitude

Modal Caustics

a0 = -410-5

f=300 MHz

Example #2: Open surface waveguide


0 km 12 km

15 km

Hei

ght [

m]

Normalized Field Strength

f=300 MHz

100 NM 10-6 contribution

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.

Conclusions and Discussions

ieee ap-ursi ohio meeting sevgi, akleman, felsen field and network theories in ece theories in ece...

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