numerical electromagnetics & semiconductor industrial applications
DESCRIPTION
National Central University Department of Mathematics. Numerical ElectroMagnetics & Semiconductor Industrial Applications. Ke-Ying Su Ph.D. 11 NUFFT & Applications. Part I : 1D-NUFFT. Outline : 1D-NUFFT. Introduction NUFFT algorithm Approach Incorporating the NUFFT into the analysis - PowerPoint PPT PresentationTRANSCRIPT
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
s
1h
h2
r1
0
r2
h3
r3
w11s
w21 21
11
2N2-1s
1N11N1-1s w
2N2w
s1N1+1
2N2+1s
a
st
h1
0
r1
h2r2
1
w1
2sw2 N
wsN+1
a
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
s
1h
h2
r1
0
r2
h3
r3
w11s
w21 21
11
2N2-1s
1N11N1-1s w
2N2w
s1N1+1
2N2+1s
a
6/67
SDA: advantages
• Easy formulation in the form of algebraic
equations
• A rigorous full-wave solution for uniform planar structures
st
h1
0
r1
h2r2
1
w1
2sw2 N
wsN+1
a
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
j
f
f
iksN
Nkkj efd
12/
2/
for j = 1, 2, …, Nd, sj [-, ], and sj's are nonuniform; Nd Nf
Idea: approximate each eiksj in terms of values
at the nearest q + 1 equispaced nodes
S jsj S j+1S j-1 Sj+q/2S j-q/2
where Sj+t = (vj+t) 2/mNf
vj = [sjmNf /2]
fjj mNkqviq
jiks
k ese /2)12/(1
1
)(
(2.1)
(2.2)
9/67
The regular Fourier matrix
fjj mNkqviq
jiks
k ese /2)12/(1
1
)(
ffjffj
ffjffj
mNNqvimNNqvi
mNNqvimNNqvi
ee
ee
/)12/(2)2/(/)12/(2)2/(
/)2/(2)2/(/)2/(2)2/(
jf
f
jf
f
sNiN
sNiN
jq
j
e
e
s
s
)12/(12/
)2/(2/
1
1
)(
)(
for a given sj and for k = -Nf/2, …, Nf/2-1
Ar(sj) = v(sj)
r(sj) = [A*A]-1[A*v(sj)] = F-1P
where F is the regular Fourier matrix with size (q+1)2
where A : Nf(q+1)
(2.2)
(2.3) closed forms
10/67
Choose k = cos(k/mNf)
nN
n
e
ee
F
f
mN
ni
m
ni
m
ni
n f
,
,
12
Closed forms
f
j
f
j
mN
qi
j
mN
qi
j
j
e
m
q
i
e
m
q
isP 122322
1
2
122sin
1
2
322sin
)(
j = sjmNf/2 vj where
(2.4)
(2.5)
(2.6)
11/67
The q+1 nonzero coefficients.
fjj mNkqviq
jkiks ese /2)12/(
1
1
1 )(
The coefficients r
1 2 3 4 5 6 7 8 9-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l
rl(
s j)
sj = 0.5207
q = 8
S jsj S j+1S j-1 Sj+q/2S j-q/2
where Sj+t = (vj+t) 2/mNf
(2.2)
r(sj) = F-1P
12/67
1
1
12/
2/
/2)12/(1)()(q N
Nk
mNkqvikkjj
k
k
fjefsd
j
f
f
iksN
Nkkj efd
12/
2/
fjj mNkqviq
jkiks ese /2)12/(
1
1
1 )(
1D-NUFFT
+
FFT
Sjsj Sj+1Sj-1 Sj+q/2Sj-q/2
(2.1)
(2.2)
(2.7)
13/67
III. Approach
The spectral domain electric fields
),(~
),(~
),(~
nGnGnG reijijij
)(~
)(),(~
nFCnG ijijij where : propagation constant i, j = z or x
s
1h
h2
r1
0
r2
h3
r3
w11s
w21 21
11
2N2-1s
1N11N1-1s w
2N2w
s1N1+1
2N2+1s
a
y
x
)(~
)(~
),(~
),(~
),(~
),(~
),(~
),(~
nJ
nJ
nGnG
nGnG
nE
nE
x
z
xxxz
zxzz
x
z
The spectral domain Green’s functions
Asymptotic extraction technique
(2.8)
14/67
0)(~
nzx nF
nh
xxnh
zxnh
zznnn enFenFenF 222 )(
~ ,)(
~ ,)(
~ 01
If the observation points and the source points are
• at the same interface :
• at different interfaces :
n =
n/a)
-1.25 -0.75 -0.25 0.25 0.75 1.25
x 104
-60
-50
-40
-30
-20
-10
0
10
n
Gzz
-1.25 -0.75 -0.25 0.25 0.75 1.25
x 104
-90
-60
-30
0
30
60
90
n
Gzx
-1.25 -0.75 -0.25 0.25 0.75 1.25
x 104
-500
0
500
1000
1500
2000
2500
n
Gxx
1)(
~ nzz nF nxx nF )(~
Asymptotic parts
(2.9)
(2.10)
15/67
Current basis functions
N
i
N
p i
ipipz
b
X
XTaxJ
1
1
021
)()(
N
i
N
pipiipx
b
XUXbxJ1
1
0
2 )(1)(
.otherwise,0
22,
)(2 ii
ii
i
i
i
wxx
wx
w
xx
X
-1 -0.5 0 0.5 1-1
-0.5
0
0.5
1 T (x)
p = 0p = 1p = 2p = 3p = 4
p
-1 -0.5 0 0.5 1-4
-3
-2
-1
0
1
2
3
4
5 U (x)
p = 0p = 1p = 2p = 3p = 4
p
Chebyshev polynomials Bessel functions
transform
(2.11a)
(2.11b)
(2.11c)
16/67
Space domain E
Expansion E-field
),(
),(
),(
),(
),(
),(
xE
xE
xE
xE
xE
xErex
rez
x
z
x
z
s
1h
h2
r1
0
r2
h3
r3
w11s
w21 21
11
2N2-1s
1N11N1-1s w
2N2w
s1N1+1
2N2+1s
a
y
x
transformSpectral domain
E~
xi
n x
z
xxxz
zxzz
x
z nenJ
nJ
nGnG
nGnG
xE
xE
)(
~)(
~
),(~
),(~
),(~
),(~
),(
),(
),(~
),(~
),(~
nGnGnG reijijij
(2.12a)
17/67
i p xxipxxipxzipxzip
zxipzxipzzipzzip
x
z
xECbxECa
xECbxECa
xE
xE
)()()()(
)()()()(
),(
),(
xi
n x
z
rexx
rexz
rezx
rezz
rex
rez ne
nJ
nJ
nGnG
nGnG
xE
xE
)(~
)(~
),(~
),(~
),(~
),(~
),(
),(
Expansion E-field
if the observation fields and the currents are
at the same interface :
at different interfaces :
n
xxjinpxtip
inewBxE )()2/()( n
xxj
n
inpztip
inewB
xE )()2/()(
where t = z or xclosed forms
numerical calculations
(2.12b)
(2.12c)
(2.13)
18/67
Unknown coefficients aip’s and bip’s
strip i i
ipz dx
X
XTxE 0
1
)(),(
2
strip i
ipix dxXUXxE 0)(1),( 2
Galerkin’s procedure
for i = 1, …, N, and p = 0, 1, …, Nb – 1
a matrix of 2NNb2NNb
(2.14a)
(2.14b)
19/67
IV. Incorporating the NUFFT into the analysis
)()()(1
1
01
1
0
xEbxEaxEN
i
N
pzxipip
N
i
N
pzzipipz
bb
N
i
N
pxxipip
N
i
N
pxzipipx
bb
xEbxEaxE1
1
01
1
0
)()()(
s
1h
h2
r1
0
r2
h3
r3
w11s
w21 21
11
2N2-1s
1N11N1-1s w
2N2w
s1N1+1
2N2+1s
a
y
x
xi
n x
z
xxxz
zxzz
x
z nenJ
nJ
nGnG
nGnG
xE
xE
)(
~)(
~
),(~
),(~
),(~
),(~
),(
),(
(2.15)
20/67
Gauss-Chebyshev quadrature
1
0
)()(gN
qjqjqztip XTdxE
1
0
)()(gN
qjqjqxtip XUhxE
where t = z or x
1
0
1
12
cos)(2
1
)()(2 gN
kkjkztip
gj
j
jqztipjq qxE
NdX
X
XTxEd
1 ..., 1, 0,,2
12 gg
k NkN
k
where xjk = xj + (wj/2)cosk
Let
Then
The advantage of NUFFT
(2.16)
(2.17)
21/67
Number of operations for MoM
the traditional SDA : Ns(2NNb)2
the proposed method : 2NNb[mNflog2(mNf)]
r3
r2
r1
0
h
h1
2
h3
s1 w1 w2s 2 NN-1s w sN+1
a
NUFFT
xi
n x
z
xxxz
zxzz
x
z nenJ
nJ
nGnG
nGnG
xE
xE
)(
~)(
~
),(~
),(~
),(~
),(~
),(
),(
22/67
Finite metallization thickness
Mixed spectral domain approach (MSDA)
st
h1
0
r1
h2r2
1
w1
2sw2 N
wsN+1
a
MN+2,b1
MN+2,b
Mb
M0,t1
11
Mt1
2MN+2,b
0
Mb
2M0,t
Mt
22
2
N+2N+1 h2
Mb
N+1M0,t h1
N+1
N+1
N+1Mt
t
23/67r1 = r2 = 8.2, r3 = 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
V. Numerical ResultsValidity Check
Table 2.1Convergence 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.
0
h1
h2
r1
r2
1s
a
sw1 22w 8s w8 9s
(Nb = 4 ) The result of HFSS is 2.6061 (33 seconds).
24/67
Table 2.2Validity 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.
0
h
h
2
1
h3
r2
r1
r3
1s
a
sw 2Validity Check
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.
25/67
Modal Propagation Characteristics
10 12.5 15 17.5 20 22.5 25-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Frequency (GHz)
/k
/k
h = 1.5 mmh = 1.27 mm
[16]2
00
2
2h = 1.1 mm
r
a
a
r
w
w s w
h
d
d
h
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.
26/67
1 5 10 15 202.14
2.165
2.19
2.215
2.24
2.6
2.45
2.3
2.15
2.75
Frequency (GHz)
/k
(
)
1
2
3
4
56
87
/k ( )
0
0
a
r
w1 s1 w2 s7 w8
h2
h1
Eight-Line Structure
Structural parameters are identical to those of Table 2.1.
27/67
-2 -1 0 1 20
5
10
15x 104
x (mm)
J z1(x
)/I 1
-2 -1 0 1 2-1.5
-1
-0.5
0
0.5
1
1.5x 105
x (mm)
J z2(x
)/I 1
-2 -1 0 1 2-1.5
-1
-0.5
0
0.5
1
1.5x 105
x (mm)
J z3(x
)/I 1
-2 -1 0 1 2-1.5
-1
-0.5
0
0.5
1
1.5x 105
x (mm)
J z4(x
)/I 1
-2 -1 0 1 2-3
-2
-1
0
1
2
3x 105
x (mm)
J z5(x
)/I 1
-2 -1 0 1 2-4
-3
-2
-1
0
1
2
3
4x 105
x (mm)
J z6(x
)/I 1
-2 -1 0 1 2-8
-6
-4
-2
0
2
4
6
8x 105
x (mm)
J z7(x
)/I 1
-2 -1 0 1 2-4
-3
-2
-1
0
1
2
3
4x 106
x (mm)
J z8(x
)/I 1
Eight-Line Structure : normalized currents at 10 GHz
1 2 3 4
5 6 7 8
28/67
0 4 8 12 16 201.55
1.65
1.75
1.85
1.95
2.05
2.15
Frequency (GHz)
/k
0
2.16
2.155
2.15
2.145
2.14
2.135
2.13
/k
0
0 4 8 12 16 20
Frequency (GHz)
s = 0.4mms = 0.3mms = 0.2mms = 0.1mm
s = 0.1mm
s = 0.4mms = 0.3mms = 0.2mm
mode 1
mode 2
mode 3
mode 4
Suspended Four-Line Structure : mode 1(odd) & 2 (even)
0
h2
3h
r2
s1 w1w s 2 s4w
a
52
1h r1
r3
h1 = h2 = 1 mm, h3 = 18 mm, r1 = r3 = 1, r2 = 8.2, w1 = w2 = w3 = w4 = 0.2 mm, s1 = s5 = 10 mm, s2 = s3 = s4 = s.
29/67
0 4 8 12 16 201.55
1.65
1.75
1.85
1.95
2.05
2.15
Frequency (GHz)
/k
0
2.16
2.155
2.15
2.145
2.14
2.135
2.13
/k
0
0 4 8 12 16 20
Frequency (GHz)
s = 0.4mms = 0.3mms = 0.2mms = 0.1mm
s = 0.1mm
s = 0.4mms = 0.3mms = 0.2mm
mode 1
mode 2
mode 3
mode 4
Suspended Four-Line Structure : mode 3 (odd) & 4 (even)
0
h2
3h
r2
s1 w1w s 2 s4w
a
52
1h r1
r3
30/67
1 5 10 15 202.1
2.375
2.65
2.925
3.2/
k
Frequency (GHz)
21
43
5
6
7
8
0
Dual-level Eight-Line Structure
11
h
h
h r11
0
2 r2 s11 w
3
21
r3
s 21w1413s w
23s 24w
a
15s25s
r1 = 10.2, r2 = 8.2, r3 = 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.
31/67
Dual-level Two-Line Structure
0 20 40 60 80 1002.7
2.9
3.1
3.3
3.5
/k
Frequency (GHz)
h = 0.127 mmh = 0.0635 mm
Mode 1
Mode 2
[17]h = 0.1905 mm0
2
2
2
w
r11h0
h2
h321
r2s
r3
a
22s21
s11 11w s12
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.
32/67
Coupled lines with finite metallization
5 10 15 20 256
7
8
9
10
11
Frequency (GHz)
Effe
cti v
e D
iele
c tri c
Co
n sta
ntMeasurement [18]
t/h = 0.01, 0.044, 0.08,1
Even mode
Odd mode
0.122, 0.18, 0.25
h
h
ts2s
0
1
1
r1
2 r21w 2w
a
s3
r1 = 12.5, r2 = 1, w1 = w2 = s2, h1 = 0.6 mm, h2 = 10 mm, and s1 = s3 = 6 mm.
33/67
Table 2.3Convergence 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.
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.