1. Antenna Modeling Using FDTD SURE Program 2004 Clemson University Michael Frye Faculty Advisor: Dr. Anthony Martin

2. Presentation Outline General Finite-Difference Time-Domain Method (FDTD) Modeling Approach Formulation of Antenna Model in FDTD Dipole Driving-Point Impedance Comparison Future Work

3. What is FDTD? Numerical technique  Computer based (computationally intensive)  Time-domain solution Modeling of electromagnetic phenomenon  Radiation, scattering, etc.

4. One FDTD Application Specific absorption rate distribution of 1,900MHz cell phone held against tilted head model Comp. Electrodynamics Taflove and Hagness

5. 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

6. Simple Finite-Difference Example Exact Value FD Approximation (Central-difference) (Forward-difference) (Reverse-difference)

7. 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

8. 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

9. 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

10. 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

11. 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”

12. 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

13. 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

14. 3D FDTD Grid Truncated by UPML

15. 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

16. 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

17. Implementation of wire in FDTD grid Components set to zero Components which decay as 1/r

18. 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

19. 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

20. Visual Results Dipole ( l=2m, a=0.005m ) radiating into a 3D FDTD grid terminated by UPML, pulse excitation

21. 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?

22. Driving-Point Impedance Comparison Antenna excited with wideband voltage source  Differentiated Gaussian Pulse chosen Known spectrum, zero DC content

23. 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

24. Comparison Results

26. 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

27. Acknowledgments Dr. Anthony Martin Chaitanya Sreerama Dr. Daniel Noneaker Dr. Xiao-Bang Xu

28. Thank You