Numerical ElectroMagnetics & Semiconductor Industrial Applications

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.