1

Complex Images

k’

k”

k0

-k0

branchcut

k0

pole

C1

C0

from the Sommerfeld identity, the complex exponentials must be a function of kz0 and therefore, we need to consider the

integration path in the complex kz0 plane

kz0’

kz0” k0

t = 0

t = T0k0T0

C1

C0

pole

00

00z Tt0,T

t1jtkk

uniform sampling is required

2

Complex Images

tBN

1ii

kbN

1ii1

i0zi eAea)k(F

tBN

1i

'i

kbN

1i

'i2

'i0z

'i eAea)k(F

)jT1/(TBii

00ieAa )jT1(k

TBb

00

0ii

3

Complete Spatial Green’s Functions

aCIaSW0aa GGGG

qCIqSW0qq GGGG

ci

rjkN

1ii

0aCI r

ea

4G

ci0

2i

2ci )jb(r

'r

e'a

4G

ci

'rjkN

1ii

0qCI

ci0

2

i2

ci )'jb('r

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 T0 and number of sampling points on

Contour C1

The restriction on uniform sampling of Contour C1 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

'dsr

1

S 'dsrn

S

6

Treatment of 1/r Singularity

'dsr

1'ds

r

1

r

e

SS

rjk 0

x

y

-

1 Triangle 1

the first term is numerically integrable, the second term can be done analytically

7

Treatment of 1/r Singularity

dxrdrd

r

x

21

21

cos

'

0 cos

1'

1

sind

sin1

1'xd

cos

cos'xd

cos

1'x

2

1

2

1

2

122

sin)

sin1

2/1

sin1

2/1('sin

sin1

1' 2

121 2

dxdx

2

1

2

1

|sin1

sin1n

2

'xsind

sin1

1

sin1

1

2

'x

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 N2 memory storage and N3 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.1221-1228.

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

AjE inc

S a )r(J)r,r(GSd )r(A

S q )r(J)r,r(GSd )r(

dkk)k(H)k(G~

)(G )2(0q,aq,a

12

SM/CG Methodthe 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

]Z[ s

]Z[ w

][Z ][Z wi

K

0i

w

V I ]Z[ ]Z[ 0w0

s

n

K

1i

wi1n

w0

s I ]Z[ V I ]Z[ ]Z[

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

q,aG~

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

dkk )k(J)k(G~

2)(G0 0

eq,a

eq,a

0J

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

dkk)k(J)k(G~

)(G0

ee

klnu lnv

du)uv(H)u(F)v(G

)e(G~

e)u(F ueu )e(Ge)v(G vev )e(Je)u(H uu

17

Fast-Hankel Transform

)nu

(P)n(F)u(F*

])nm[(H)n(F)m(G **

where G* is the approximation of G

the filter coefficient function is defined as)v(H*

du)uv(H)

u(P)v(H*

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

)uasinh(

)usin(a)u(P

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 DECN

DECN

10ln

for an optimized filter function, the smoothing parameter a and the sampling interval satisfy the equation 4a

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

v

F)m(G*

20

Accuracy of FHT

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

0.001 0.01 0.1 1 10Distance in wavelengths

Am

plit

ude

of

Ga

FHTNI

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

0.001 0.01 0.1 1 10

Distance in wavelengths

Am

plit

ude

of

Gq

FHTNI

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

)mln

(P)m(G1

)(G *eq,a

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

N

1iminiiq,a SexpR)(G

min0

23

Accuracy of Matrix Pencil Curve Fitting

-3

-2

-1

0

1

2

0.01 0.1 1 10Distance in wavelengths

Imag

inar

y pa

rt o

f Ga

FHT

MP

-2

0

2

4

6

0.01 0.1 1 10Distance in wavelengths

Real p

art

of G

a

FHT

MP

24

Accuracy of Matrix Pencil Curve Fitting

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.01 0.1 1 10

Distance in wavelengths

Imagin

ary

part

of G

q FHTMP

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0.01 0.1 1 10Distance in wavelengths

Real p

art

of

Gq

FHTMP

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