1. Limitation of Pulse Basis/Delta Testing Discretization: TE-Wave EFIE difficulties lie in the behavior of fields produced by the pulse expansion functions consider the scattered electric field due to a current source extended from x o to x 1 and oriented in the x direction

2. Further Explanation

3. Discussions electric field produced by the constant pulse of current density J x (x) is singular at the edges of the source segment due to the presence of line charges associated with the discontinuity in current density at the end of the segment as fictitious line charges give rise to infinite tangential electric field at the two edges of every cell in the model point matching is done at the center so all the matrix elements are finite and solution can be found, but accuracy may be affected

4. Discussions for MFIE with TE polarization, although the transverse electric field is singular, the magnetic field H z produced by the pulse is finite along the source segment and therefore the solution is fine for EFIE with TM polarization, there is no fictitious line charges generated as and therefore the solution is also fine to avoid generation of fictitious line charges for the EFIE TE case, a smoother basis function is required for formulation with one derivative, we can use pulse basis and delta testing for formulation with two derivatives, we need to use triangular basis with delta or pulse testing

5. TE Wave Scattering from PEC Strips or Cylinders – EFIE we have studied MFIE for TE polarization but MFIE cannot treat thin structure since is not the same as where the magnetic field on one size is equal to and zero on the other when the equivalent principle is applied

6. Use of EFIE EFIE can be used for thin structures but its implementation is more difficult it is advisable to use basis and testing functions having additional degrees of differentiability to compensate for additional derivatives presence in the TE EFIE we will consider the use of subsectional triangular basis functions with pulse testing which together provide two degrees of differentiability beyond that of the pulse/delta combination

7. Formulation of EFIE total tangential E equals to zero, i.e., tangential scattered E is equal and opposite to tangential incident E magnetic vector potential electric scalar potential continuity equation

8. Triangular Basis Function triangular basis function spans over two current segments for a close structure, we have M basis functions for a M- segment structure for an open structure, we only have M-1 basis function two pulses are also used to approximate the triangular basis function t t n-1 tntn t n+1

9. Pulse Doublet corresponds to the slopes of the triangular edges jnjn j n /(t n -t n-1 ) -j n /(t n+1 -t n ) area = j n pulse doublet cancel each other, zero total charge no fictitious line charge generated

10. Testing Function the choice of pulse testing function permits the analytical treatment of the gradient operator appearing in the mixed potential form of the EFIE according to the equation that

11. Matrix Equation

12. Bistatic Cross Section further approximate the first integral by approximating the triangle with two pulses and delta testing the equation the scalar potential contribution dominates when R is made small bistatic cross section is given by

13. TM Wave Scattering from Inhomogeneous Dielectric Cylinders – Volume EFIE Discretization with Pulse Basis and Delta Testing Functions convenient to convert the original scattering problem into an equivalent form more amenable to a direct solution replace the inhomogeneous dielectric scatterer with equivalent induced polarization currents and charges radiating in free space x y EiEi  r  x,y)        outside scatterer inside scatterer

14. Formulation of the Volume Integral Equation the scatterer is now replaced by induced currents and charges radiating in free space For TM case, we have E z, H x and H y components only and therefore, no induced charge

15. Volume Integral Equation P n (x,y)=1 if (x,y) with cell n 0 otherwise J z (x,y)=

16. Final Equation

17. Approximation to Matrix Evaluation approximate the square cell with a circular area of the same surface area so that the integral can be evaluated analytically

18. Approximation to Matrix Evaluation note that when and only appears in the diagonal term

19. Bistatic Cross Section note that the sampling rate should be 10/ d where

20. Scattering from Homogeneous Dielectric Cylinders: Surface Integral Equations Discretized with Pulse Basis and Delta Testing Functions, TM Case x y S 00 E 2, H 2 E 1, H 1 11 for an inhomogeneous cylinder, we employed the volume integral equation formulation in which a volume discretization is required for a homogeneous cylinder, it is more convenient to formulate the problem using the surface integral equation approach in which only a surface discretization is necessary

21. Formulation of the Surface Equivalent Problem 1 x y PEC E 2, H 2     0 J 2,M 2 radiating in free space

22. Formulation of the Surface Equivalent Problem 2 radiating in a homogeneous medium of x y PEC E 1, H 1     0 J 1,M 1

23. Surface Equivalent Problem 1 x y PEC E 2, H 2     0 J 2,M 2 for TM case

24. Surface Equivalent Problem 1 x y PEC E 2, H 2     0 J 2,M 2

25. Surface Equivalent Problem 2 x y PEC E 1, H 1     0 J 1,M 1 no source

26. Surface Equivalent Problem 2 x y E 1, H 1     0 J 1,M 1

27. Matrix Equation Using Puse/Delta Functions

28. Matrix Elements Using Puse/Delta Functions

29. For the Self Terms recall that 0.5-0.5

30. Combined Field Equation for PEC Cylinder x y HiHi PEC ii H z i =exp(-jk[x cos  i + y sin  i ]) 1 0 circumference, pulse basis/delta testing EFIE MFIE

31. Combined Field Equation for PEC Cylinder For the EFIE, the solution will not be unique if has a nonzero solution. Let us consider the problem of finding the resonant frequencies of a cavity by setting or [A]{x}=0 to determine the eigenvalues of matrix A from which the resonant frequencies can be calculated. The current is then equal to a linear combination of the eigenvectors correspond to each of the eigenvalues and therefore

32. Uniqueness of the Solution Does this produce an external field? The answer is no since (no source). For the magnetic field equation,, just inside S. If is not zero then but which implies that and hence, the field outside S is not unique.

33. Combined Field Equation for PEC Cylinder then the solution is unique on the PEC

34. Uniqueness of the Solution = 0 the last term represent real power flowing inside S which is equal to 0 for lossless media and > 0 for lossy media (sourceless) just inside Son S

35. Uniqueness of the Solution If  is real and positive, on S +, otherwise the above expression cannot be zero. Therefore. The solution to the combined field equation is unique at all frequencies. The combined field equation requires the same number of unknowns but the matrix elements are more complicated to evaluate. Typical value of  is between 0.2 and 1.