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

3/67 Part I : 1D-NUFFT

4/67 Outline : 1D-NUFFT 1.Introduction 2.NUFFT algorithm 3.Approach 4.Incorporating the NUFFT into the analysis 5.Results and discussions 6.Conclusion

5/67 I. Introduction Finite difference time domain approach (FDTD) Finite element method (FEM) Numerical methods Spectral domain approach (SDA) Singular integral equation (SIE) Electric-field integral equation (EFIE) Analytical formulations

6/67 SDA: advantages Easy formulation in the form of algebraic equations A rigorous full-wave solution for uniform planar structures

7/67 1) Green’s function : The spectral series has a poor convergence Asymptotic Extraction Technique SDA: disadvantages and solutions 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  

8/67 II. NUFFT Algorithm for j = 1, 2, …, N d, s j  [- ,  ], and s j 's are nonuniform; N d  N f Idea: approximate each e iks j in terms of values at the nearest q + 1 equispaced nodes where S j+t = (v j +t) 2  /mN f v j = [s j mN f /2  ] (2.1) (2.2)

9/67 The regular Fourier matrix for a given s j and for k = -N f /2, …, N f /2-1 Ar(s j ) = v(s j ) r(s j ) = [A * A] -1 [A * v(s j )] = F -1 P where F is the regular Fourier matrix with size ( q +1) 2 where A : N f  (q+1)    (2.2) (2.3) closed forms

10/67 Choose  k = cos(k  /mN f ) Closed forms  j = s j mN f /2   v j where (2.4) (2.5) (2.6)

11/67 The q+1 nonzero coefficients. The coefficients r where S j+t = (v j +t) 2  /mN f (2.2) r(s j ) = F -1 P

12/67 1D-NUFFT + FFT  (2.1) (2.2) (2.7)

13/67 III. Approach The spectral domain electric fields where  : propagation constant i, j = z or x The spectral domain Green’s functions Asymptotic extraction technique (2.8)

14/67 If the observation points and the source points are at the same interface : at different interfaces :  n = n  /a) Asymptotic parts (2.9) (2.10)

15/67 Current basis functions Chebyshev polynomialsBessel functions transform (2.11a) (2.11b) (2.11c)

16/67 Space domain E Expansion E- field transformSpectral domain (2.12a)

17/67 Expansion E- field if the observation fields and the currents are at the same interface : at different interfaces : where t = z or x closed forms numerical calculations (2.12b) (2.12c) (2.13)

18/67 Unknown coefficients a ip ’ s and b ip ’ s Galerkin’s procedure for i = 1, …, N, and p = 0, 1, …, N b – 1 a matrix of 2N N b  2N N b  (2.14a) (2.14b)

19/67 IV. Incorporating the NUFFT into the analysis (2.15)

20/67 Gauss-Chebyshev quadrature where t = z or x where x jk = x j + (w j /2)cos  k Let Then The advantage of NUFFT (2.16) (2.17)

21/67 Number of operations for MoM the traditional SDA : N s (2N N b ) 2 the proposed method : 2N N b [mN f log 2 (mN f )] NUFFT

22/67 Finite metallization thickness Mixed spectral domain approach (MSDA)

23/67  r1 =  r2 = 8.2,  r3 = 1, a = 40, h 1 + h 2 = 1.8, h 3 = 5.4, w 1 through w 8 be 0.26, 0.22, 0.18, 0.14, 0.16, 0.2, 0.24 and 0.28, and s 1 through s 9 be , 0.25, 0.21, 0.17, 0.15, 0.19, 0.23, 0.27, and All dimensions are in mm 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. (N b = 4 )The result of HFSS is (33 seconds).

24/67 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, s 1 = s 2 = 8.1, h 1 = h 2 = 1.8 and h 3 = 5.4, all in mm. Validity Check  r1 =  r3 = 1,  r2 = 8.2, a = 18, w = 1.8, s 1 = s 2 = 8.1, h 1 = h 2 = 1.8 and h 3 = 5.4, all in mm.

25/67 Modal Propagation Characteristics Single-Line Structure a = 12.7 mm, w = 1.27 mm, h 1 = 0 mm, h 3 = mm, s 1 = s 2 = (a – w)/2,  r =

26/67 Eight-Line Structure Structural parameters are identical to those of Table 2.1.

27/67 Eight-Line Structure : normalized currents at 10 GHz

28/67 Suspended Four-Line Structure : mode 1 (odd) & 2 (even) h 1 = h 2 = 1 mm, h 3 = 18 mm,  r1 =  r3 = 1,  r2 = 8.2, w 1 = w 2 = w 3 = w 4 = 0.2 mm, s 1 = s 5 = 10 mm, s 2 = s 3 = s 4 = s.

29/67 Suspended Four-Line Structure : mode 3 (odd) & 4 (even)

30/67 Dual-level Eight-Line Structure  r1 = 10.2,  r2 = 8.2,  r3 = 1, a = 40, h 1 = 1.27, h 2 = 0.53, h 3 = 5.4, w 11 through w 14 are 0.22, 0.14, 0.2, and 0.28, w 21 through w 24 are 0.26, 0.18, 0.16, and 0.24, s 11 through s 15 are , 0.56, 0.5, 0.74, and , and s 21 through s 25 are , 0.68, 0.46, 0.62, and All dimensions are in mm.

31/67 Dual-level Two-Line Structure a = 25.4 mm, h 1 = w 11 = w 21 = mm, h 3 = mm, s 11 = s 22 = mm, s 12 = s 21 = mm,  r1 =  r2 = 12, and  r3 = 1.

32/67 Coupled lines with finite metallization  r1 = 12.5,  r2 = 1, w 1 = w 2 = s 2, h 1 = 0.6 mm, h 2 = 10 mm, and s 1 = s 3 = 6 mm.

33/67 Table 2.3 Convergence Analysis and Comparison of the CPU time for an Odd Mode of a Pair of Coupled Lines with t/h 1 = 0.01 Obtained by the MSDA and the Proposed Method. Coupled lines with finite metallization 5 GHz

34/67 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.