1. Analysis of Free Space Radiation Using Finite Difference Time Domain Method
Diyu Yang
Mentor: Xu Chen Advisor: José E. Schutt-Ainé
Abstract
Implementing Finite Difference Time Domain (FDTD) solver for radiation problem from an infinitely long current source.
Analyzing the phase error of the finite difference time domain method.
FDTD Method in Computational Electromagnetism
Problem Set up
Finite Difference Formulas
The first and second derivative of a function f(x) can be approximated by the following equations:
Now let’s consider a two-dimensional electromagnetic problem with a source ,Maxwell’s equations can be reduced to
Using Yee’s Finite Difference Time Domain Method, we can rewrite the partial differential equations above as
where i and j denote the x and y-direction spatial indices, and n denotes the time step index.
Computational domain: 10cm × 10cm rectangular box, with i=j=200 grid points in each dimension. The dimension of each grid is dx=dy=0.05cm
Gaussian current Pulse in the middle of the box:
Lossless, free space, source frequency = 1.0e9 Hz
Time domain cut: t=n∙dt, with n=300, dt=0.5ps.
Assume perfect electrical conductor outside the computational domain, then the boundary value of the computation domain is set to the Dirichlet condition:
Stability condition:
(Figures cited from reference 2)
Phase error analysis
The numerical error of FDTD can be calculated by taking the percentage difference between the analytical value of propagation index k and the numerical value obtained by FDTD.
This error is different at different radiation direction φ. Our goal is to minimize this error difference with respect to φ.
Formula for phase error:
Take dt=a∙dx/c, where c is the speed of light: plot for different values of a:
Note that according to stability condition, the value of a should be less than Plot for phase error with respect to dt
Result of Radiation
Screenshot of the radiation at t=50, 150, 200 and 300ps:
We can observe from the result that the wave bounces back after encountering the boundary of the rectangular box due to our Dirichlet boundary condition.
(Potential research topic in the future: Absorbing the boundary condition)
Acknowledgements:
Professor José E. Schutt-Ainé
My mentor: Xu Chen
References
‘Theory and Computation of Electromagnetic Fields’; Jin, J.; 2010
‘Analysis of Free Space and Aperture Radiation Using Finite Difference Time Domain Method’; Xu Chen; 2013