1 EEE 431 Computational Methods in Electrodynamics Lecture 9 By Dr. Rasime Uyguroglu

2 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

3 One dimensional free space formulation: Assume a plane wave with the electric filed having Ex, magnetic field having Hy components and traveling in the z direction.

4 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Maxwell’s Equations become:

5 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Taking central difference approximation for both temporal and spectral derivatives:

6 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD)

7 Notice that: n means a time t= Everything is discretized for writing a computer code. The calculations are interleaved in both time and space. For example the new value of Ex is calculated from the previous value of Ex and the most recent values of Hy.

8 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Writing the expressions of Ex and Hy in C computer code: ex[k]=ex[k]-(dt/(eps0*dz))*(hy[k]-hy[k-1]) hy[k]=hy[k]-(dt/(mu0*dz))*(ex[k+1]-ex[k]) Note that n,n+1/2, n-1/2 superscripts are ignored. Also note that k+1/2 and k-1/2 are rounded off in order to specify a position in an array in the program

9 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Writing a C program code: Calculate Ex field by using a loop. Calculate the source. i.e the initial condition. Generate a Gaussian pulse in the center of the problem space. Apply the B.C. to find the Ex values at the boundaries of the problem space. Calculate the Hy fields by using a loop.

10 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Generation of the Gaussian pulse: pulse=exp(-((n*dt-t0)^2)/(t1^2)) ex[ks]=pulse

11 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) The stability criteria: Where n is the dimension number of the simulation.

12 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) For one dimensional simulation: Choose This is not the best choice but, it is simple.

13 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) If a wave is traveling towards a boundary in free space, it is traveling at c, the velocity of light. So in one time step of FDTD algorithm, it travels a distance:

14 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Then an acceptable boundary condition may be: Store a value of Ex(1) for two time steps and then put it in Ex(0).

15 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) B.C.in C computer code assume that the domain ends at k=KE:

16 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Now, write a complete C program to simulate a one-dimensional wave propagation. Assume that the source is at the center of the problem space. Observe the wave propagation. Then modify your program by applying the source at k=3. Observe the wave propagation.

17 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Observe the wave propagation w.r.t. the time and space. Change the number of the time steps and observe the propagation. Change the stability factor from 0.5 to 1, 1.1 and What happens?

18 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Exercises: 1)Modify your program to have two sources. One at kc-20and kc+20 where kc is the center of the problem space. 2)Instead of Ex source use Hy source at at k=kc. 3)Modify you program for a dielectric material. Plot for each case the wave propagation w.r.t. the time and space.

19 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) 4)Modify your program for a problem which is partly free space and partly dielectric.