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