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Numerical Methods in Modeling a Dipole Above Earth

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Numerical Methods in Modeling a Dipole Above Earth Powered By Docstoc
					Modeling a Dipole Above
Earth

   Saikat Bhadra
   Advisor : Dr. Xiao-Bang Xu
   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
        Radiating Above an Impedance Surface
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
   Integral Advantages
       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

				
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posted:3/15/2012
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