Numerical ElectroMagnetics & Semiconductor Industrial Applications

1. National Central University Department of Mathematics Numerical ElectroMagnetics& Semiconductor Industrial Applications Ke-Ying Su Ph.D. 11 NUFFT & Applications

3. Part I : 1D-NUFFT

4. Outline : 1D-NUFFT Introduction NUFFT algorithm Approach Incorporating the NUFFT into the analysis Results and discussions Conclusion

5. I. Introduction Numerical methods • Finite difference time • domain approach (FDTD) • Finite element method • (FEM) Analytical formulations • Spectral domain approach (SDA) • Singular integral equation (SIE) • Electric-field integral equation (EFIE)

6. SDA: advantages • Easy formulation in the form of algebraic • equations • A rigorous full-wave solution for uniform • planar structures

7. SDA: disadvantages and solutions 1) Green’s function : The spectral series has a poor convergence Asymptotic Extraction Technique  2) Galerkin’s procedure: Both of Green’s function and Electric field are in space domain or spectral dimain SDA :number of operations N2 Green’s function is in spectral domain and Electric field is in space domain NUFFT :number of operations N 

8. II. NUFFT Algorithm forj = 1, 2, …, Nd, sj  [-, ], andsj's are nonuniform; Nd  Nf (2.1) Idea: approximate eacheiksjin terms of values at the nearestq + 1 equispaced nodes (2.2) where Sj+t = (vj+t) 2/mNf vj = [sjmNf /2]

9. closed forms The regular Fourier matrix for a given sj and fork = -Nf/2, …, Nf/2-1 (2.2)   Ar(sj) = v(sj) whereA : Nf(q+1) r(sj) = [A*A]-1[A*v(sj)]= F-1P  (2.3) where F is the regular Fourier matrix with size (q+1)2

10. Closed forms (2.4) Choosek = cos(k/mNf) (2.5) where j = sjmNf/2-vj (2.6)

11. The coefficients r (2.2) r(sj) = F-1P where Sj+t = (vj+t) 2/mNf The q+1 nonzero coefficients.

12. FFT 1D-NUFFT (2.1) + (2.2)  (2.7)

13. III. Approach The spectral domain electric fields The spectral domain Green’s functions Asymptotic extraction technique (2.8) whereb : propagation constant i, j= z or x

14. Asymptotic parts If the observation points and the source points are • at the same interface : (an = np/a) (2.9) • at different interfaces : (2.10)

15. Current basis functions (2.11a) (2.11b) (2.11c) Chebyshev polynomials Bessel functions transform

16. Spectral domain Expansion E-field transform Space domain E (2.12a)

17. closed forms Expansion E-field (2.12b) (2.12c) if the observation fields and the currents are at the same interface : (2.13) where t = z or x at different interfaces : numerical calculations

18. Unknown coefficients aip’s and bip’s Galerkin’s procedure (2.14a) (2.14b) for i = 1, …, N, and p = 0, 1, …, Nb – 1  a matrix of 2NNb2NNb

19. IV. Incorporating the NUFFT into the analysis (2.15)

20. The advantage of NUFFT Gauss-Chebyshev quadrature Let (2.16) where t = z or x Then (2.17) where xjk = xj + (wj/2)cosqk

21. NUFFT Number of operations for MoM the traditional SDA : Ns(2NNb)2 the proposed method : 2NNb[mNflog2(mNf)]

22. Finite metallization thickness Mixed spectral domain approach (MSDA)

23. V. Numerical Results Validity Check Table 2.1 Convergence Analysis and Comparison of the CPU time for a Quasi-TEM Mode of an Eight-Line Microstrip Structure Obtained by the Traditional SDA and the Proposed Method. (Nb = 4) The result of HFSS is 2.6061 (33 seconds). er1 = er2 = 8.2, er3 = 1, a = 40, h1 + h2 = 1.8, h3 = 5.4, w1 through w8 be 0.26, 0.22, 0.18, 0.14, 0.16, 0.2, 0.24 and 0.28, and s1 through s9 be 18.495, 0.25, 0.21, 0.17, 0.15, 0.19, 0.23, 0.27, and 18.355. All dimensions are in mm

24. Validity Check Table 2.2 Validity Check of the Modal Solutions Obtained by the Proposed Method. Structure in Fig.1(a):r1 = r3 = 1, r2 = 8.2, a = 18, w = 1.8, s1 = s2 = 8.1, h1 = h2 = 1.8 and h3 = 5.4, all in mm. r1 = r3 = 1, r2 = 8.2, a = 18, w = 1.8, s1 = s2 = 8.1, h1 = h2 = 1.8and h3 = 5.4, all in mm.

25. Modal Propagation Characteristics Single-Line Structure a = 12.7 mm, w = 1.27 mm, h1 = 0 mm, h3 = 11.43 mm, s1 = s2 = (a – w)/2, r = 8.875.

26. Eight-Line Structure Structural parameters are identical to those of Table 2.1.

27. Eight-Line Structure : normalized currents at 10 GHz 1 2 3 4 5 6 7 8

28. Suspended Four-Line Structure : mode 1(odd) & 2 (even) h1 = h2 = 1 mm, h3 = 18 mm, er1 = er3 = 1, er2 = 8.2, w1 = w2 = w3 = w4 = 0.2 mm, s1 = s5 = 10 mm, s2 = s3 = s4 = s.

29. Suspended Four-Line Structure : mode 3 (odd) & 4 (even)

30. Dual-level Eight-Line Structure er1 = 10.2, er2 = 8.2, er3 = 1, a = 40, h1 = 1.27, h2 = 0.53, h3 = 5.4, w11 through w14 are 0.22, 0.14, 0.2, and 0.28, w21 through w24 are 0.26, 0.18, 0.16, and 0.24, s11 through s15 are 19.005, 0.56, 0.5, 0.74, and 18.355, and s21 through s25 are 18.495, 0.68, 0.46, 0.62, and 18.905. All dimensions are in mm.

31. Dual-level Two-Line Structure a = 25.4 mm, h1 = w11 = w21 = 0.127 mm, h3 = 25.146 mm, s11 = s22 = 12.895 mm, s12 = s21 = 12.378 mm, r1 =r2 = 12, andr3 = 1.

32. Coupled lines with finite metallization r1 = 12.5, r2 = 1, w1 = w2 = s2, h1 = 0.6 mm, h2 = 10 mm, and s1 = s3 = 6 mm.

33. Coupled lines with finite metallization : @ 5 GHz Table 2.3 Convergence Analysis and Comparison of the CPU time for an Odd Mode of a Pair of Coupled Lines with t/h1 = 0.01 Obtained by the MSDA and the Proposed Method.

34. VI. Conclusion • NUFFT and asymptotic extraction technique are • used to enhance the computation. • Very high efficiency is obtained for shielded • single and multiple coupled microstrips. • The results have good convergence. • Mode solutions with varying substrate heights, • microstrips at different dielectric interfaces or • finite metallization thickness are investigated • and presented.