Antenna Modeling Using FDTD

SURE Program 2004Clemson University

Michael FryeFaculty 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

First-order derivatives (time and space) replaced with finite-difference approximations

“Update equations” developed for calculation of field values in a discrete 3D grid

1

t

H

E1

t

E

H

Simple Finite-Difference Example Exact Value

FD Approximation

(Central-difference)

(Forward-difference)

(Reverse-difference)

lim0( ) ( )( )

xc abx

ƒ ƒƒ

( ) ( )( ) c abx

ƒ ƒƒ

Development of update equations Consider Ex component equation of Ampere’s Law

Simply problem by reducing to 2D (for illustration)

Choose to evaluate at time: t=n and location: x=i, y=j

1Ex Hz Hy

t y z

1Ex Hz

t y

, ,

,

| |1

|

n ni j i j

i j

Ex Hz

t y

Development of update equations Approx. time derivative with central-difference

Resulting expression (Ex and Hz displaced in time)

Hz evaluated at integer time-steps Ex evaluated at integer +/- ½ time-steps

1/ 2 1/ 2, , ,| | |n n ni j i j i jEx Ex Ex

t t

1/ 2 1/ 2, , ,

,

| | |1

|

n n ni j i j i j

i j

Ex Ex Hz

t y

Development of update equations Approx. spatial partial derivative with central-difference

Result: Ex and Hz also displaced in space

Hz evaluated at integer +/- ½ y points along grid Ex evaluated at integer y points along grid

, 1 2 , 1 2, | || n nni j i ji j Hz HzHz

y y

1/ 2 1/ 2, 1 2 , 1 2, ,

,

| || | 1

|

n nn ni j i ji j i j

i j

Hz HzEx Ex

t y

Development of update equations Resulting update equation for Ex for 2D case

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

1/ 2 1/ 2, , , 1 2 , 1 2

,

1| | | |

|n n n ni j i j i j i j

i j

tEx Ex Hz Hz

y

Yee Cell (Typically Used for FDTD)

Basis of 3D computational grid Builds lattice of Yee cells

Field components displaced in space and time

E and H field locations interlocked in space

Solution is “time-stepped”

Antenna model in FDTD

Basic elements for FDTD antenna model Open region

Infinite computational grid Contains antenna, modeled structures, etc.

Representation of antenna structure in FDTD grid Thin-wire model (one example)

Voltage feed Provides antenna excitation

Uniaxial Perfectly Matched Layer Problem: FDTD grid cannot be “infinite”

Implies unlimited computational time and resources

Solution: Truncate with conductive material layer 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

UPML region

Free space 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

Components set to zero Components set to zero

Components which decay as 1/r

Faraday’s Law contour path model

Faraday’s Law

Applied to contour C and surface S

New update equations derived for circulating H components

Yee grid illustrates both differential and integral forms

1

C S

dd d

t

H

E l s

y

x

z

Ez

0y

Ey

Hx

Thin wire

C

a

Ey

Ez

S

y y

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

Antenna and EM Modeling with MATLAB, Sergey N. Makarov

Method of Moments patch code (freq. domain) Dipole Driving Point Impedance compared

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

Voltage and current calculated for each time step Transients allowed to “die-out”

Discrete Fourier Transform 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