Antenna Modeling Using FDTD
SURE Program 2004
Clemson University
Michael Frye
Faculty Advisor: Dr. Anthony Martin
Presentation Outline

General Finite-Difference Time-Domain Method
(FDTD) Modeling Approach

Formulation of Antenna Model in FDTD

Dipole Driving-Point Impedance Comparison

Future Work
What is FDTD?

Numerical technique



Computer based (computationally intensive)
Time-domain solution
Modeling of electromagnetic phenomenon

Radiation, scattering, etc.
One FDTD Application

Specific absorption rate
distribution of 1,900MHz
cell phone held against
tilted head model
Comp.
Electrodynamics
Taflove and Hagness
FDTD Modeling Approach

Approximation of Maxwell’s Curl Equations

Faraday’s Law and Ampere’s Law

Differential, time-domain form
H
1
  E
t



E 1
 H
t 
First-order derivatives (time and space) replaced with
finite-difference approximations
“Update equations” developed for calculation of field
values in a discrete 3D grid
Simple Finite-Difference Example
ƒ(x)
c
Exact Value

ƒ
(
c
)

ƒ
(
a
)

ƒ(b)  lim


x
x  0






FD Approximation
a
b
x

ƒ
(
c
)

ƒ
(
a
)

ƒ(b) 


x






(Central-difference)
x(Forward-difference)
(Reverse-difference)
Development of update equations

Consider Ex component equation of Ampere’s Law
Ex 1  Hz Hy 
 

t
  y
z 

Simply problem by reducing to 2D (for illustration)
Ex 1  Hz 
 
t
  y 

Choose to evaluate at time: t=n and location: x=i, y=j
Ex |in, j
t
n



Hz
|
1
i, j



 |i , j  y 
Development of update equations

Approx. time derivative with central-difference
Ex |in, j
t


Ex |in,j1/ 2  Ex |in,j1/ 2
t
Resulting expression (Ex and Hz displaced in time)
Ex |in,j1/ 2  Ex |in,j1/ 2
t


n



Hz
|
1
i, j



 |i , j  y 
Hz evaluated at integer time-steps
Ex evaluated at integer +/- ½ time-steps
Development of update equations

Approx. spatial partial derivative with central-difference
Hz |in, j
y


Hz |in, j 1 2  Hz |in, j 1 2
y
Result: Ex and Hz also displaced in space
Ex |in,j1/ 2  Ex |in,j1/ 2
t


n
n


Hz
|

Hz
|
1
i , j 1 2
i , j 1 2



 |i , j 
y

Hz evaluated at integer +/- ½ y points along grid
Ex evaluated at integer y points along grid
Development of update equations

Resulting update equation for Ex for 2D case
n 1/ 2
i, j
Ex |
n 1/ 2
i, j
 Ex |
1

 |i , j
 t 
n
n
Hz
|

Hz
|
i , j 1 2
i , j 1 2 
 
 y 

Fully explicit solution for each Ex point on grid
 Only information at previous time steps needed
 No matrix inversion needed (Implicit solution)
 Introduces stability issues (Courant condition)
 Species maximum ratio of spatial and time step

Remaining update equations derived similarly
 Faraday’s law provides H component update equations
Yee Cell (Typically Used for FDTD)
z
Hy

Ez
Hx
Hz
Hx
Hy
Hz
Hz



Ey

Ex
Basis of 3D computational grid
Field components displaced in
space and time
E and H field locations
interlocked in space
Solution is “time-stepped”
Hx
Hy
y
x
Builds lattice of Yee cells
Antenna model in FDTD

Basic elements for FDTD antenna model

Open region



Representation of antenna structure in FDTD grid


Infinite computational grid
Contains antenna, modeled structures, etc.
Thin-wire model (one example)
Voltage feed

Provides antenna excitation
Uniaxial Perfectly Matched Layer

Problem: FDTD grid cannot be “infinite”


Solution: Truncate with conductive material layer



Implies unlimited computational time and resources
Similar to walls in anechoic chamber
Allows antennas to be simulated as radiating into open space
with a finite FDTD grid
Desired characteristics



Reflectionless boundary regardless of incident field polarization
or angle
Incident fields attenuated to zero (through conductivity)
Reasonably small addition to computational grid
3D FDTD Grid Truncated by UPML
The PML on the Top
The PML on the Left
PEC wall
Free space region
UPML region
Thin-wire FDTD model

Consider modeling a very thin wire

Needed for dipole, monopole, etc.

Option 1: Decrease cell size fit wire into cell




Diameter of wire equals cell width
Significantly increases computation time
Cubic approx. of circular cross-section
Option 2: Use sub-cellular modeling techniques


Modeled features can be smaller than FDTD grid size
Cell size independent of wire radius
Faraday’s Law contour path model

Uses integral form of Faraday’s Law


Results not obvious from differential FD approach
Special update equations developed


Affects field components immediately around wire
Near-field physics behavior built into field values
immediately around wire


Tangential E set to zero (along wire)
Circulating H and radial E fields decay as 1/r
 Radial distance away from center of wire
Implementation of wire in FDTD grid
a
z
z
Ez
Ey
Ex
Hy
Ey
Hz
Hx
Ez
a
z
Hx
Ez
Ey
Hx
Hx
Hy
Hz
Ey
Hz
Ez
Ey
Ex
Hx
yH
Hy
Ex
Ez
Hy
Ex
Ez
Ex
y
x
x
Ez

x
Components
set to zero

Components which decay as 1/r
y
y
Faraday’s Law contour path model
a
z
Faraday’s Law
Thin wire
1
Ey
Hx
Ez
C
S

Ez

Ey
C

x
y0
y y
y
dH
ds
S t
 E dl   
Applied to contour C and
surface S
New update equations
derived for circulating H
components
Yee grid illustrates both
differential and integral
forms
Antenna Feeding

Gap-feed method


Provides problem excitation
Relates incident voltage to E-field in feeding gap


Added to tangential E-Field component
Shows very little dependence on grid size


Acts like infinitesimal feed gap
Important for consistent results
Visual Results

Dipole ( l=2m, a=0.005m ) radiating into a 3D FDTD grid
terminated by UPML, pulse excitation
Driving-Point Impedance Comparison

Need quantitative verification of FDTD model



Dipole Driving Point Impedance compared



Antenna and EM Modeling with MATLAB, Sergey N.
Makarov
Method of Moments patch code (freq. domain)
Dipole parameters: length 2m, radius 0.005m
Frequency range: 25MHz-500MHz
How can freq. information be determined from
time-domain results?
Driving-Point Impedance Comparison

Antenna excited with wideband voltage source

Differentiated Gaussian Pulse chosen

Known spectrum, zero DC content
Driving-Point Impedance Comparison

Energy radiates into grid



Discrete Fourier Transform


Voltage and current calculated for each time step
Transients allowed to “die-out”
Compare directly to frequency information
FDTD Solution convergence



Spatial cell size dictated by desired frequencies
 10 or more cells per wavelength
Computation time increases as spatial size decreases
Finer grids typically result in higher accuracy
Comparison Results
Comparison Results
Future Work

Development of Near Field to Far Field
transformation





Currently in progress
FDTD intrinsically Near Field technique
Radiation patterns
Wideband Far Zone information
Design/analysis of reconfigurable antennas

Nonlinear switching devices
Acknowledgments

Dr. Anthony Martin

Chaitanya Sreerama

Dr. Daniel Noneaker

Dr. Xiao-Bang Xu
Thank You