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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