Modeling a Dipole Above Earth Saikat Bhadra Advisor : Dr. Xiao-Bang Xu Clemson SURE 2005
Dec 26, 2015
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
Inhomogeneous Materials
Time Varying
Non-flat & non-Euclidean surfaces
Location : Austin, TX
Layered Materials
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
( , , )x y zObservation Point
Dipole( ', ', ')x y z
Impedance Half Space
Free 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 :
Separate diffracted and direct components Reflection Coefficients transformed using
Laplace transform
Bessel function identities
EIT Formulation
'2 2 2 2
02 2 0
ˆ ˆ ˆ( , ') 24 '
ikRikRdv z
iZ I e eE r r l x y z e d
k x z y z x y R R
EIT Solution – Diffracted Fields
'( ) ( ') ( ') ( )R x x y y z z i ( ') ( ') ( ')R x x y y z z
( , , )x y zObservation Point
Dipole
( ', ', ')x y z
Impedance Half Space
Free Space
( , )
( , , )
Direct
Diffracted
EIT Solution Integral Advantages
Rapidly Decays Non-Oscillatory Easy numerical evaluation after exchange of integration and
differentiation
Exact Image Theory
'2 2 2 2
02 2 0
ˆ ˆ ˆ( , ') 24 '
ikRikRdv z
iZ I e eE r r l x y z e d
k x z y z x y R R
'( ) ( ') ( ') ( ' )R x x y y z z i
( ') ( ') ( ')R x x y y z z
( , , )x y zObservation Point
Dipole( ', ', ')x y z
Impedance Surface
Free Space ( , )
( , , )
Direct
Diffracted
Dipole( ', ', ')x y zDipole( ', ', ' )x y z i Dipole( ', ', ' )x y z i Dipole( ', ', ' )x y z i Dipole( ', ', ' )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 :
Current distribution assumed to sinusoidal Transmission line approximation Inaccurate when dipole comes close to half space
2
21
1 l h
in z z
h
Z I z E z dzI
Numerical Techniques
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
'2 2 2 2
02 2 0
ˆ ˆ ˆ( , ') 24 '
ikRikRdv z
iZ I e eE r r l x y z e d
k x z y z x y R R
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