1 Complex Images k’k’ k”k” k0k0 -k0-k0 branch cut   k 0 pole C1C1 C0C0 from the Sommerfeld identity, the complex exponentials must be a function of k z0 and therefore, we need to consider the integration path in the complex k z0 plane k z0 ’ k z0 ” k0k0 t = 0 t = T 0 k0T0k0T0 C1C1 C0C0 pole uniform sampling is required

2 Complex Images

3 Complete Spatial Green’s Functions

4 Discussions the success of the complex image method highly depends on the use of either the Prony’s method or matrix pencil method to compute the complex exponential series which depends on the number of complex image terms, the truncation value of T 0 and number of sampling points on Contour C 1 The restriction on uniform sampling of Contour C 1 also limits the accuracy of the method Successful implementation of the method also depends on the implementation of the Prony’s method and matrix pencil method

5 Singularities when the separation between the basis and testing functions is small, the scalar potential will dominate as the scalar potential contribution dominates the self-term of the impedance matrix, we can approximate the current density to have a constant value within the plate self-term with contributions from the vector and scalar potentials has the following singularities

6 Treatment of 1/r Singularity x y --  1 Triangle 1 the first term is numerically integrable, the second term can be done analytically

7 Treatment of 1/r Singularity

8 in the previous lecture, we discuss the MPIE modeling of microstrip structure we discuss the complex image method which allows efficient implementation of the spatial Green’s function leading to fast matrix fill time as solving a full matrix requires N 2 memory storage and N 3 operations, we need a different matrix solver when N is large MoM Solution of Microstrip Structure

9 majority of the plate interactions is far sampling of the Green’s function from centroid of one plate to the centroid of another weighted by the area of the source plate can approximate the integral if the centroids of all the plates fall on a uniformly spaced grid, we can compute the interactions efficiently using the FFT what if these centroids do not fall on a uniform grid A Sparse-Matrix/Canonical Grid Method for Densely-Packed Interconnects Refer to MTT-49,No.7, pp

10 for a large matrix equation, we cannot store the whole matrix as it requires too much memory therefore, solution based on matrix inversion is not possible the large matrix is solved by iterative method in an iterative solution, we need to perform matrix-vector multiplications repeatedly the computational complexity and the memory requirement are reduced to O(NlnN) and O(N) respectively in SM/CG method Iterative Solution to Large Matrix Equation

11 MPIE FormulatioSM/CG Method

12 SM/CG Method the impedance matrix is decomposed into the sum of a sparse matrix, denoting the strong neighborhood interactions, and a dense matrix, denoting the weak far-interactions through a Taylor series expansion, we have, the iterative procedure is given by due to the translationally invariable kernels in the Green’s functions, the weak-matrix vector multiplication can be efficiently performed via the FFT’s

13 Close-Form Spatial-Domain Green’s Functions from the FHT when using the FHT algorithm to calculate the Sommerfeld integral, the integral is reduced to a discrete convolution and the result is the response of a Hankel filter before applying the FHT, the real poles of must be found and extracted since in the FHT method, the integration path is along the real axis the contributions of these poles can be calculated by residue calculus

14 Close-Form Spatial-Domain Green’s Functions from the FHT after extracting the poles and some quasi-static terms, we have where is the zero’th-order Bessel function this integral can be performed numerically using the fast Hankel transform algorithm which are discrete data this discrete data will be curved fitted so that a closed-form expression can be obtained

15 Fast-Hankel Transform in the FHT algorithm, the spectral-domain Green’s function is sampled exponentially, which means that the sample will be very dense for small k  the Green’s function may have sharp peaks and fast changes when k  is small in spectral domain, which maps to the far- field region in the spatial domain compared with the CIM, in which the sampling is uniform, the dense sampling in the FHT algorithm for small range can grasp the fast changes and therefore can provide more robust and accurate results for the far-field region in the spatial domain

16 Fast-Hankel Transform

17 Fast-Hankel Transform where G* is the approximation of G the filter coefficient function is defined as

18 Fast-Hankel Transform the integral is computed as a contour integral on the complex plane and its expression can be derived as a sum of residues P(u) is a interpolating function and a is a smoothing parameter the sampling interval  is usually determined by the number of sampling points per decade for an optimized filter function, the smoothing parameter a and the sampling interval satisfy the equation

19 Discussions Hankel filters constructed in such a way have attractive features the error decreases exponentially with the cut-off frequency, which means that even a moderate increase in sampling density will make the error decrease drastically the filter coefficient function has explicit series representations, and the coefficients decrease exponentially as, which makes it possible to evaluate them to any desired accuracy only a limited number of sampling values of are needed to obtain accurately converged at each sampling point

20 Accuracy of FHT

21 Analytical Expression to obtain analytical expressions of spatial Green's functions from the numerical results of the FHT, we approximate them by a sum of complex exponentials using the well-known matrix pencil method in the matrix pencil method, the sampling points are required to be uniform, although the direct results of the FHT are exponentially sampled to obtain a uniform sampled sequence, we apply the same interpolating function used in the FHT algorithm

22 Matrix Pencil Curve Fitting the surface-wave and quasi-dynamic contributions are combined together with the FHT data to obtain the whole spatial analytical Green’s function by applying the matrix pencil method the expression as a sum of complex exponentials is we can simply use the quasi-dynamic contributions to approximate the Green’s function for

23 Accuracy of Matrix Pencil Curve Fitting

24 Accuracy of Matrix Pencil Curve Fitting

25 Far Interaction Calculation if the ratio of the maximum side of the two interacting triangles to the separation of their centroids is below 20%, a point-to- point evaluation of the Green's function weighted by the areas of the triangles is sufficient efficient evaluation of the far-interaction contributions in the MVM is reduced to efficient convolution between the Green’s function and the current vector