1. 1 EEE 431 Computational Methods in Electrodynamics Lecture 13 By Dr. Rasime Uyguroglu Rasime.uyguroglu@emu.edu.tr

2. 2 FINITE DIFFERENCE TIME DOMAIN METHOD (FDTD) Lossy Material

3. 3 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) When a material has a finite conductivity, a conduction current term is added to Ampere’s Law (different than the source term). Thus:

4. 4 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) As before assuming the x component of E and the variation only in the z direction:

5. 5 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) This equation can be expanded about a point : to find FDTD update equation. However, when loss is present, the undifferentiated electric field appears on the left side of the equation.

6. 6 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) With the assumption that the electric field point is, there is no electric field at this space-time point. This problem can be solved by using the average of the electric field to the either side of the desired point:

7. 7 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Thus a suitable discretization of Ampere’s Law when loss is present is:

8. 8 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) As before this can be solved for, which the present field, in term of purely past fields. The result is:

9. 9 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Now, consider the Faraday’s Law: When, no magnetic loss is assumed

10. 10 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Now considering that there is, varying with z direction:

11. 11 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Discretized form:

12. 12 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Solving for Yields the following update equation:

13. 13 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) Consider a lossy dielectric half-space which starts at node 100. The relative permittivity is 9 as before. However there is a also an electric loss present such that :

14. 14 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) A previously employed simple ABC does not work at the right edge of the grid. It can be removed. But the left side of the grid can be terminated by using the same ABC as before.

15. 15 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Material) The magnetic field update remained the same. That us observe the pulse propagation for different time steps. (i.e. multiples of 10)

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31. 31 FINITE DIFFERENCE TIME DOMAIN METHOD ( Lossy Material)

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34. 34 FINITE DIFFERENCE TIME DOMAIN METHOD (Lossy Dielectric)

35. 35 FINITE DIFFERENCE TIME DOMAIN METHOD (TFSF-Lossy Material) Conclusion: The pulse decays as it propagates in the lossy region and eventually decays to a rather negligible value. The lack of an ABC at the right side of the grid is not really a concern in this particular instance.