# Numerical Methods in Modeling a Dipole Above Earth

Document Sample

```					Modeling a Dipole Above
Earth

Clemson SURE 2005
Overview

   Objective
   Problem Background & Theory
   Results
   Problems in the EIT Model
   Concluding Remarks
Objective

   Accurate modeling of a dipole
   Linear Antenna
   Lossy Earth
   Material Properties
   Scientific Model
   K. Sarabandi, M. D. Casciato, and I. Koh
Efficient Calculation of the Fields of A Dipole
Solving Electromagnetic Problems
   The Emag Bible : Maxwell’s Equations
   Available in integral and differential forms

   Vector Potential
   Links Magnetic and Electric Fields
Antenna Environment

Non-flat & non-Euclidean surfaces
Time Varying

Inhomogeneous Materials

Layered Materials

Location : Austin, TX
Simplifications

   Simplify math and assume :
   Flat Earth Model
   Two Layers
   Upper half space – “air”
   Lower half space – lossy earth
   Euclidean (rectangular)
geometry
   Infinitesimal Vertical Dipole
 Superposition to extend to
finite dipoles
Electric Field
   In this type of problem, two fields are involved
   Direct Electric Fields
     Fields due to antenna radiating
     Solution in closed form & well documented
   Diffracted Electric Fields
     Fields from antenna that are reflecting off the lower
surfaces
     Subject of research since 1909
Observation Point
( x, y, z )
Dipole
( x ', y ', z ')                                         Free Space
(  ,  )

Impedance Half Space   ( ,  ,  )
Original Solution – Diffracted Fields

   Arnold Sommerfeld (1909)
   Sommerfeld Integrals
   Non-analytic
   Numerical integration difficult
   Requires asymptotic techniques
   Valid for certain regions
   Convergence difficult
Original Solution – Diffracted Fields
cont’d
Exact Image Theory Solution
   Sarabandi, Casciato, Koh (2002)
   Source Equation :
EIT Formulation
   Separate diffracted and direct components
   Reflection Coefficients transformed using
Laplace transform

   Bessel function identities
EIT Solution – Diffracted Fields
ikR ' 
iZ 0 I   2
       2       2  2    eikR
              
  e

E (r , r ')  
d
lz    ˆ
x       ˆ   2  2  z 
y             ˆ          2  e                  d 
4 k  xz     yz      x y    R                   R '   
v
                           
0

R '( )  ( x  x ')  ( y  y ')  ( z  z  i )         R  ( x  x ')  ( y  y ')  ( z  z ')

Direct                       Observation Point
( x, y, z )
Dipole
Free Space
( x ', y ', z ')                           Diffracted
(  ,  )

Impedance Half Space ( , , )
EIT Solution
   Rapidly Decays
   Non-Oscillatory
   Easy numerical evaluation after exchange of integration and
differentiation
Exact Image Theory
ikR ' 
iZ 0 I   2
       2       2  2    eikR
              
  e

E (r , r ')  
d
lz    x
ˆ       y   2  2  z 
ˆ             ˆ          2  e                  d 
4 k  xz     yz      x y    R                   R '   
v
                           
0

R '( )  ( x  x ')  ( y  y ')  ( z  z ' i )

R  ( x  x ')  ( y  y ')  ( z  z ')

Direct                  Observation Point
( x, y, z )
Dipole
( x ', y ', z ')                                                             Free Space
Diffracted                      (  ,  )

Impedance Surface       ( ,  ,  )

Dipole
( xDipole
', y ',  z ')
Dipole
( x ', y ',  z ' i )
( x ', y ',Dipole
 z ' i )
( x ', y ',Dipole
 z ' i )
( x ', y ',  z ' i )
Finite Length Dipoles
   Sarabandi’s model uses infinitesimal dipole
   Finite dipole can be approximated by a sum of
infinitesimal dipoles
   Superposition Principle
Calculating Input Impedance

   Induced EMF Method :
2l  h
1
Zin   2
I1     I  z  E  z  dz
h
z    z

   Current distribution assumed to sinusoidal
   Transmission line approximation
   Inaccurate when dipole comes close to half space
Numerical Techniques
ikR ' 

iZ 0 I   2     2       2  2    eikR
              
  e

E (r , r ')  
d
lz    x
ˆ       y   2  2  z 
ˆ             ˆ          2  e                  d 
4 k  xz     yz      x y    R                   R '   
v
                           
0

   Gaussian Integration
   Useful in many emag problems
   Handles singular integrands better
   More accurate than rectangular, trapezoidal, and
Simpson’s rule
   Integral Truncation
   Can’t numerically evaluate an infinite integral
   Vectorized Code
Results

   Computational time varies with antenna
location
   Frequency independence
   Asymptotically approaches original antenna
impedance
Results cont’d
Problems of the EIT Model

   Recall the breakdown of electric field into
diffracted and direct components
   Diffracted fields should go to zero if the half-
space is removed
   There is no longer any surface for waves to
bounce off of
   Numerical Results disagree
   Currently finding theoretical errors of the
model
Problems of the EIT Model cont’d
Concluding Remarks
   EIT model could be promising but problems
need to be solved
   Research Applications
   Antenna Design
   Integral Equations & Numerical Methods
Future Work

   Solve the EIT model problems
   Extend the problem to dipoles of arbitrary
orientation
   Develop more accurate model of current
distribution
   Investigate different source models
Acknowledgments

   Dr. Xu
   Dr. Noneaker
Questions?
Environmental Variables
   Time varying
   Inhomogeneous Materials (x,y)
   Water
   Grass
   Concrete
   Layered Materials (z)
   Trees, Grass, Soil
   Non-flat surfaces
   Amorphous (non-Euclidean) geometries
   Mutual Coupling

```
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
 views: 3 posted: 3/15/2012 language: English pages: 26